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G = C22×S32order 144 = 24·32

Direct product of C22, S3 and S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C22×S32, C32⋊C24, C625C22, C3⋊S3⋊C23, (C2×C6)⋊8D6, (C3×C6)⋊C23, (C3×S3)⋊C23, C31(S3×C23), C61(C22×S3), (S3×C6)⋊10C22, (S3×C2×C6)⋊7C2, (C22×C3⋊S3)⋊6C2, (C2×C3⋊S3)⋊8C22, SmallGroup(144,192)

Series: Derived Chief Lower central Upper central

C1C32 — C22×S32
C1C3C32C3×S3S32C2×S32 — C22×S32
C32 — C22×S32
C1C22

Generators and relations for C22×S32
 G = < a,b,c,d,e,f | a2=b2=c3=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 912 in 284 conjugacy classes, 104 normal (6 characteristic)
C1, C2 [×3], C2 [×12], C3 [×2], C3, C22, C22 [×34], S3 [×8], S3 [×12], C6 [×6], C6 [×11], C23 [×15], C32, D6 [×12], D6 [×50], C2×C6 [×2], C2×C6 [×13], C24, C3×S3 [×8], C3⋊S3 [×4], C3×C6 [×3], C22×S3 [×2], C22×S3 [×27], C22×C6 [×2], S32 [×16], S3×C6 [×12], C2×C3⋊S3 [×6], C62, S3×C23 [×2], C2×S32 [×12], S3×C2×C6 [×2], C22×C3⋊S3, C22×S32
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], S32, S3×C23 [×2], C2×S32 [×3], C22×S32

Permutation representations of C22×S32
On 24 points - transitive group 24T232
Generators in S24
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 22)(2 24)(3 23)(4 20)(5 19)(6 21)(7 17)(8 16)(9 18)(10 14)(11 13)(12 15)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 22)(2 23)(3 24)(4 21)(5 19)(6 20)(7 18)(8 16)(9 17)(10 15)(11 13)(12 14)

G:=sub<Sym(24)| (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,20)(5,19)(6,21)(7,17)(8,16)(9,18)(10,14)(11,13)(12,15), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,23)(3,24)(4,21)(5,19)(6,20)(7,18)(8,16)(9,17)(10,15)(11,13)(12,14)>;

G:=Group( (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,20)(5,19)(6,21)(7,17)(8,16)(9,18)(10,14)(11,13)(12,15), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,23)(3,24)(4,21)(5,19)(6,20)(7,18)(8,16)(9,17)(10,15)(11,13)(12,14) );

G=PermutationGroup([(1,8),(2,9),(3,7),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,22),(2,24),(3,23),(4,20),(5,19),(6,21),(7,17),(8,16),(9,18),(10,14),(11,13),(12,15)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,22),(2,23),(3,24),(4,21),(5,19),(6,20),(7,18),(8,16),(9,17),(10,15),(11,13),(12,14)])

G:=TransitiveGroup(24,232);

C22×S32 is a maximal subgroup of
C62.91C23  D64D12  D65D12  C62.125C23  D6≀C2
C22×S32 is a maximal quotient of
D12.33D6  D12.34D6  D1223D6  D1224D6  D1227D6  Dic6.24D6  Dic612D6  D1212D6  D1213D6  D12.25D6  Dic6.26D6  D1215D6  D1216D6  C32⋊2+ 1+4

36 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O3A3B3C6A···6F6G6H6I6J···6Q
order12222···222223336···66666···6
size11113···399992242···24446···6

36 irreducible representations

dim111122244
type+++++++++
imageC1C2C2C2S3D6D6S32C2×S32
kernelC22×S32C2×S32S3×C2×C6C22×C3⋊S3C22×S3D6C2×C6C22C2
# reps11221212213

Matrix representation of C22×S32 in GL6(ℤ)

-100000
0-10000
001000
000100
000010
000001
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
0000-11
0000-10
,
100000
010000
001000
000100
000001
000010
,
-110000
-100000
00-1100
00-1000
000010
000001
,
010000
100000
000-100
00-1000
000010
000001

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22×S32 in GAP, Magma, Sage, TeX

C_2^2\times S_3^2
% in TeX

G:=Group("C2^2xS3^2");
// GroupNames label

G:=SmallGroup(144,192);
// by ID

G=gap.SmallGroup(144,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,490,3461]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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