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## G = C4×S32order 144 = 24·32

### Direct product of C4, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×S32
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32
 Lower central C32 — C4×S32
 Upper central C1 — C4

Generators and relations for C4×S32
G = < a,b,c,d,e | a4=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 352 in 116 conjugacy classes, 46 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C4×S32
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C2×S32, C4×S32

Permutation representations of C4×S32
On 24 points - transitive group 24T224
Generators in S24
(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 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 24)(2 21)(3 22)(4 23)(5 20)(6 17)(7 18)(8 19)(9 14)(10 15)(11 16)(12 13)

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

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

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

G:=TransitiveGroup(24,224);

C4×S32 is a maximal subgroup of
S32⋊C8  C24⋊D6  S32⋊Q8  S32⋊D4  D1223D6  Dic612D6  D1215D6  Dic36S32
C4×S32 is a maximal quotient of
C24⋊D6  C24.63D6  C24.64D6  C24.D6  C62.6C23  Dic35Dic6  C62.8C23  C62.47C23  C62.48C23  C62.49C23  Dic34D12  C62.51C23  C62.53C23  C62.72C23  C62.74C23  C62.91C23  Dic36S32

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 size 1 1 3 3 3 3 9 9 2 2 4 1 1 3 3 3 3 9 9 2 2 4 6 6 6 6 2 2 2 2 4 4 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 C4×S3 S32 C2×S32 C4×S32 kernel C4×S32 S3×Dic3 C6.D6 S3×C12 C4×C3⋊S3 C2×S32 S32 C4×S3 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 2 1 2 1 1 8 2 2 2 2 8 1 1 2

Matrix representation of C4×S32 in GL4(𝔽5) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 3 0 0 4 3 0 0 3 3 4 3 0 4 3 0
,
 1 2 2 3 1 3 3 1 0 4 0 1 1 3 4 1
,
 3 0 0 2 0 4 2 1 2 2 0 2 1 0 0 1
,
 4 2 3 3 4 4 1 2 0 4 0 1 4 4 2 2
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,4,3,0,3,3,3,4,0,0,4,3,0,0,3,0],[1,1,0,1,2,3,4,3,2,3,0,4,3,1,1,1],[3,0,2,1,0,4,2,0,0,2,0,0,2,1,2,1],[4,4,0,4,2,4,4,4,3,1,0,2,3,2,1,2] >;

C4×S32 in GAP, Magma, Sage, TeX

C_4\times S_3^2
% in TeX

G:=Group("C4xS3^2");
// GroupNames label

G:=SmallGroup(144,143);
// by ID

G=gap.SmallGroup(144,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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