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## G = C22×S32order 144 = 24·32

### Direct product of C22, S3 and S3

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

 Derived series C1 — C32 — C22×S32
 Chief series C1 — C3 — C32 — C3×S3 — S32 — C2×S32 — C22×S32
 Lower central C32 — C22×S32
 Upper central C1 — C22

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, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C32, D6, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3×C6, C22×S3, C22×S3, C22×C6, S32, S3×C6, C2×C3⋊S3, C62, S3×C23, C2×S32, S3×C2×C6, C22×C3⋊S3, C22×S32
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, S32, S3×C23, C2×S32, 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 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 3C 6A ··· 6F 6G 6H 6I 6J ··· 6Q order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 3 6 ··· 6 6 6 6 6 ··· 6 size 1 1 1 1 3 ··· 3 9 9 9 9 2 2 4 2 ··· 2 4 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D6 D6 S32 C2×S32 kernel C22×S32 C2×S32 S3×C2×C6 C22×C3⋊S3 C22×S3 D6 C2×C6 C22 C2 # reps 1 12 2 1 2 12 2 1 3

Matrix representation of C22×S32 in GL6(ℤ)

 -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

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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