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G = C4xS32order 144 = 24·32

Direct product of C4, S3 and S3

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

Aliases: C4xS32, C12:4D6, D6.8D6, Dic3:5D6, (S3xC12):8C2, (C3xC12):5C22, (S3xDic3):6C2, C6.D6:5C2, C6.7(C22xS3), (C3xC6).7C23, C32:1(C22xC4), (S3xC6).8C22, C3:Dic3:2C22, (C3xDic3):5C22, C3:1(S3xC2xC4), C2.1(C2xS32), (C4xC3:S3):7C2, C3:S3:1(C2xC4), (C2xS32).2C2, (C3xS3):1(C2xC4), (C2xC3:S3).14C22, SmallGroup(144,143)

Series: Derived Chief Lower central Upper central

C1C32 — C4xS32
C1C3C32C3xC6S3xC6C2xS32 — C4xS32
C32 — C4xS32
C1C4

Generators and relations for C4xS32
 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, C2xC4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C22xC4, C3xS3, C3:S3, C3xC6, C4xS3, C4xS3, C2xDic3, C2xC12, C22xS3, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, S3xC2xC4, S3xDic3, C6.D6, S3xC12, C4xC3:S3, C2xS32, C4xS32
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S32, S3xC2xC4, C2xS32, C4xS32

Permutation representations of C4xS32
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);

C4xS32 is a maximal subgroup of
S32:C8  C24:D6  S32:Q8  S32:D4  D12:23D6  Dic6:12D6  D12:15D6  Dic3:6S32
C4xS32 is a maximal quotient of
C24:D6  C24.63D6  C24.64D6  C24.D6  C62.6C23  Dic3:5Dic6  C62.8C23  C62.47C23  C62.48C23  C62.49C23  Dic3:4D12  C62.51C23  C62.53C23  C62.72C23  C62.74C23  C62.91C23  Dic3:6S32

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H12I12J
order1222222233344444444666666612121212121212121212
size113333992241133339922466662222446666

36 irreducible representations

dim111111122222444
type++++++++++++
imageC1C2C2C2C2C2C4S3D6D6D6C4xS3S32C2xS32C4xS32
kernelC4xS32S3xDic3C6.D6S3xC12C4xC3:S3C2xS32S32C4xS3Dic3C12D6S3C4C2C1
# reps121211822228112

Matrix representation of C4xS32 in GL4(F5) generated by

2000
0200
0020
0002
,
1300
4300
3343
0430
,
1223
1331
0401
1341
,
3002
0421
2202
1001
,
4233
4412
0401
4422
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] >;

C4xS32 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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