# Groups of small order

Compiled by John Pedersen, Dept of Mathematics, University of South Florida, jfp@math.usf.edu

## Order 1 and all prime orders (1 group: 1 abelian, 0 nonabelian)

All groups of prime order p are isomorphic to C_p, the cyclic group of order p.
A concrete realization of this group is Z_p, the integers under addition modulo p.

## Order 4 (2 groups: 2 abelian, 0 nonabelian)

• C_4, the cyclic group of order 4
• V = C_2 x C_2 (the Klein four group) = symmetries of a rectangle. A presentation for the group is
            <a, b; a^2 = b^2 = (ab)^2 = 1>
The Cayley table of the group is (putting c = ab):
              | 1  a  b  c
--+-----------
1 | 1  a  b  c
a | a  1  c  b
b | b  c  1  a
c | c  b  a  1 
A matrix representation is the four 2x2 matrices
            [1 0]   [1  0]   [-1 0]   [-1  0]
[0 1],  [0 -1],  [ 0 1],  [ 0 -1] 
A permutation representation is the following four elements of S_4:
           (1),  (1 2)(3 4),  (1 3)(2 4) and (1 4)(2 3).
Its lattice of subgroups is (in the notation of the Cayley table)
                      V
/  |  \
<a> <b> <c>
\  |  /
{1} 

## Order 6 (2 groups: 1 abelian, 1 nonabelian)

• C_6
• S_3, the symmetric group of degree 3 = all permutations on three objects, under composition. In cycle notation for permutations, its elements are (1), (1 2), (1 3), (2, 3), (1 2 3) and (1 3 2).
There are four proper subgroups of S_3; they are all cyclic. There are the three of order 2 generated by (1 2), (1 3) and (2 3), and the one of order 3 generated by (1 2 3). Only the one of order 3 is normal in S_3.
A presentation for S_3 is (where s corresponds to (1 2) and t to (2 3)):
            <s,t; s^2 = t^2 = 1, sts = tst>
Another presentation (with s <-> (1 2 3), t <-> (1 2)) is
            <s,t; s^3 = t^2 = 1, ts = s^2 t>
In terms of this second presentation, with 2 = s^2, u = ts and v = ts^2, the Cayley table is
              | 1  s  2  t  u  v
--+-----------------------
1 | 1  s  2  t  u  v
s | s  2  1  v  t  u
2 | 2  1  s  u  v  t
t | t  u  v  1  s  2
u | u  v  t  2  1  s
v | v  t  u  s  2  1 
This shows S_3 is isomorphic to D_3, the dihedral group of degree 3, that is, the symmetries of an equilateral triangle (this never happens for n > 3). The lattice of subgroups of S_3 is
                          S_3
/  /   |   \
<t> <u> <v>  <s>
\  \   |   /
{1} 
The first three proper subgroups have order two, while <s> has order three and is the only normal one.
The center of S_3 is trivial (in fact Z(S_n) is trivial for all n.)
The automorphism group of S_3 is isomorphic to S_3.

## Order 8 (5 groups: 3 abelian, 2 nonabelian)

• C_8
• C_4 x C_2
• C_2 x C_2 x C_2
• D_4, the dihedral group of degree 4, or octic group. It has a presentation
             <s, t; s^4 = t^2 = e; ts = s^3 t>
In terms of these generators (s corresponds to rotation by pi/2 and t to a reflection about an axis through a vertex), the eight elements are 1,s,s^2,s^3,t,ts,ts^2 and ts^3. Using the notation 2 = s^2, 3 = s^3, t2 = ts^2 and t3 = ts^3, the Cayley table is
              | 1  s  2  3  t ts t2 t3
--+------------------------
1 | 1  s  2  3  t ts t2 t3
s | s  2  3  1 t3  t ts t2
2 | 2  3  1  s t2 t3  t ts
3 | 3  1  s  2 ts t2 t3  t
t | t ts t2 t3  1  s  2  3
ts |ts t2 t3  t  3  1  s  2
t2 |t2 t3  t ts  2  3  1  s
t3 |t3  t ts t2  s  2  3  1 
Its subgroup lattice is
                             D_4
/           |         \
{1,s^2,t,ts^2}    <s>   {1,s^2,st,ts}
/     |         \    |   /       |      \
<ts^2>   <t>           <s^2>       <st>     <ts>
\      \            |          /        /
{1} 
Of these, the proper normal subgroups are the three of order four and <s^2> of order two.
The center of D_4 is {1,s^2}, which is also its derived group.
The automorphism group of D_4 is isomorphic to D_4.

• Q, the quaternion group. It has a presentation
             <s, t; s^4 = 1, s^2 = t^2, sts = t>
Q can be realized as consisting of the eight quaternions 1, -1, i, -i, j, -j, k, -k, where i is the imaginary square root of -1, and j and k also obey j^2 = k^2 = -1. These quaternions multiply according to clockwise movement around the figure
                               i
/      \
k  ----  j 
For example, ij = k and ji = -k (negative because anticlockwise).
A matrix representation is given by s and t in the above presentation corresponding to these two 2x2 matrices over the complex numbers:
           s = [i  0]     t = [0 i]
[0 -i]         [i 0] 
The subgroup lattice of Q is
                                  Q
/     |     \
<s>    <st>   <t>
\     |     /
<s^2>
|
{1} 
All of these subgroups are normal in Q.
The center of Q is {1,s^2}, which is also its derived group.
The automorphism group of Q is isomorphic to S_4.

• C_9
• C_3 x C_3

• C_10
• D_5

## Order 12 (5 groups: 2 abelian, 3 nonabelian)

• C_12
• C_6 x C_2
• A_4, the alternating group of degree 4, consisting of the even permutations in S_4. The subgroup lattice of A_4 is
                               A_4
/     \        \        \         \
<(12)(34),(13)(24)>    <(123)>  <(124)>  <(134)>  <(234)>
/       |       \         |       /       /         /
<(12)(34)> <(13)(24)> <(14)(23)> |      /       /         /
\       \          \      /     /       /         /
{1} 
The only proper normal subgroup is <(12)(34),(13)(24)>.
• D_6, isomorphic to S_3 x C_2 = D_3 x C_2
• T which has the presentation
       <s, t; s^6 = 1, s^3 = t^2, sts = t>
T is the semidirect product of C_3 by C_4 by the map g : C_4 -> Aut(C_3) given by g(k) = a^k, where a is the automorphism a(x) = -x.
Another presentation for T is
        <x,y; x^4 = y^3 = 1, yxy = x>
In terms of these generators, using AB for x^A y^B, the Cayley table for T is
           | 00  10  20  30  01  02  11  21  31  12  22  32
------+-----------------------------------------------
1 = 00| 00  10  20  30  01  02  11  21  31  12  22  32
x = 10| 10  20  30  00  11  12  21  31  01  22  32  02
x^2 = 20| 20  30  00  10  21  22  31  01  11  32  02  12
x^3 = 30| 30  00  10  20  31  32  01  11  21  02  12  22
y = 01| 01  12  21  32  02  00  10  22  30  11  20  31
y^2 = 02| 02  11  22  31  00  01  12  20  32  10  21  30
xy = 11| 11  22  31  02  12  10  20  32  00  21  30  01
x^2y = 21| 21  32  01  12  22  20  30  02  10  31  00  11
x^3y = 31| 31  02  11  22  32  30  00  12  20  01  10  21
xy^2 = 12| 12  21  32  01  10  11  22  30  02  20  31  00
x^2y^2 = 22| 22  31  02  11  20  21  32  00  12  30  01  10
x^3y^2 = 32| 32  01  12  21  30  31  02  10  22  00  11  20 
A 2x2 matrix representation of this group over the complex numbers is given by
                   [0  i]              [w   0 ]
x <--> [i  0]       y <--> [0  w^2]  
where i is a square root of -1 and w is nonreal cube root of 1, for example w = e^{2\pi i/3}.

• C_14
• D_7

C_15.

## Order 16 (14 groups: 5 abelian, 9 nonabelian)

• C_16
• C_8 x C_2
• C_4 x C_4
• C_4 x C_2 x C_2
• C_2 x C_2 x C_2 x C_2
• D_8
• D_4 x C_2
• Q x C_2, where Q is the quaternion group
• The quasihedral (or semihedral) group of order 16, with presentation
        <s,t; s^8 = t^2 = 1, st = ts^3>
• The modular group of order 16, with presentation
        <s,t; s^8 = t^2 = 1, st = ts^5>
The elements are s^k t^m, k = 0,1,...,7, m = 0,1.
The center is {1,s^2,s^4,s^6}.
Its subgroup lattice is
                             G
/   |   \
<s^2,t>  <s>  <st>
/   |   \  |   /
<s^4,t> <s^2t>  <s^2>
/  |   \  |     /
<t> <s^4t>  <s^4>
\  |    /
{1}            
This is the same subgroup lattice structure as for the lattice of subgroups of C_8 x C_2, although the groups are of course nonisomorphic.
The automorphism group is isomorphic to D_4 x C_2
Reference: Weinstein, Examples of Groups, pp. 120-123.
• The group with presentation
           < s,t; s^4 = t^4 = 1, st = ts^3 >
The elements are s^i t^j for i,j = 0,1,2,3.
The center of G is {1,s^2,t^2,s^2t^2}.
Reference: Weinstein, pp. 124--128.
• The group with presentation
  <a,b,c; a^4 = b^2 = c^2 = 1, cbca^2b = 1, bab = a, cac = a>
• The group G_{4,4} with presentation <s,t; s^4 = t^4 = 1, stst = 1, ts^3 = st^3 >
• The generalized quaternion group of order 16 with presentation <s,t; s^8 = 1, s^4 = t^2, sts = t >

## Order 18 (5 groups: 2 abelian, 3 nonabelian)

• C_18
• C_6 x C_3
• D_9
• S_3 x C_3
• The semidirect product of C_3 x C_3 with C_2 which has the presentation
    <x,y,z; x^2 = y^3 = z^3 = 1, yz = zy, yxy = x, zxz = x>

## Order 20 (5 groups: 2 abelian, 3 nonabelian)

• C_20
• C_10 x C_2
• D_10
• The semidirect product of C_5 by C_4 which has the presentation
       <s,t; s^4 = t^5 = 1, tst = s>
• The Frobenius group of order 20, with presentation
       <s,t; s^4 = t^5 = 1, ts = st^2>

This is the Galois group of x^5 -2 over the rationals, and can be represented as the subgroup of S_5 generated by (2 3 5 4) and (1 2 3 4 5).

## Order 21 (2 groups: 1 abelian, 1 nonabelian)

• C_21
• <a,b; a^3 = b^7 = 1, ba = ab^2> This is the Frobenius group of order 21, which can be represented as the subgroup of S_7 generated by (2 3 5)(4 7 6) and (1 2 3 4 5 6 7), and is the Galois group of x^7 - 14x^5 + 56x^3 -56x + 22 over the rationals (ref: Dummit & Foote, p.557).

• C_22
• D_11

## Order 24 (15 groups: 3 abelian, 12 nonabelian)

• C_24
• C_2 x C_12
• C_2 x C_2 x C_6
• S_4
• S_3 x C_4
• S_3 x C_2 x C_2
• D_4 x C_3
• Q x C_3
• A_4 x C_2
• T x C_2
• Five more nonabelian groups of order 24
Reference: Burnside, pp. 157--161.

• C_25
• C_5 x C_5

• C_26
• D_13

## Order 27 (5 groups: 3 abelian, 2 nonabelian)

• C_27
• C_9 x C_3
• C_3 x C_3 x C_3
• The group with presentation
        <s,t; s^9 = t^3 = 1, st = ts^4 >
• The group with presentation
  <x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx>
Reference: Burnside, p. 145.

• C_28
• C_2 x C_14
• D_14
• D_7 x C_2

## Order 30 (4 groups: 1 abelian, 3 nonabelian)

• C_30
• D_15
• D_5 x C_3
• D_3 x C_5
Reference: Dummit & Foote, pp. 183-184.

A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu