<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 1A 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}
<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 1This 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.
<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 1Its 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.
<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 ---- jFor example, ij = k and ji = -k (negative because anticlockwise).
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.
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)>.
<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.
<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 20A 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}.
<s,t; s^8 = t^2 = 1, st = ts^3>
<s,t; s^8 = t^2 = 1, st = ts^5>The elements are s^k t^m, k = 0,1,...,7, m = 0,1.
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.
< s,t; s^4 = t^4 = 1, st = ts^3 >The elements are s^i t^j for i,j = 0,1,2,3.
<a,b,c; a^4 = b^2 = c^2 = 1, cbca^2b = 1, bab = a, cac = a>
<x,y,z; x^2 = y^3 = z^3 = 1, yz = zy, yxy = x, zxz = x>
<s,t; s^4 = t^5 = 1, tst = s>
<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).
<s,t; s^9 = t^3 = 1, st = ts^4 >
<x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx>Reference: Burnside, p. 145.