## 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 isS_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 isD_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 figurei / \ 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 isQ / | \ <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.

## Order 9 (2 groups: 2 abelian, 0 nonabelian)

- C_9
- C_3 x C_3

## Order 10 (2 groups: 1 abelian, 1 nonabelian)

- 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}.

## Order 14 (2 groups: 1 abelian, 1 nonabelian)

- C_14
- D_7

## Order 15 (1 group: 1 abelian, 0 nonabelian)

- 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 isG / | \ <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).

## Order 22 (2 groups: 1 abelian, 1 nonabelian)

- 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.

## Order 25 (2 groups: 2 abelian, 0 nonabelian)

- C_25
- C_5 x C_5

## Order 26 (2 groups: 1 abelian, 1 nonabelian)

- 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.

## Order 28 (4 groups: 2 abelian, 2 nonabelian)

- 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