non-abelian, soluble, monomial
Aliases: (S32):C8, C4.16S3wrC2, (C3xC12).16D4, C3:Dic3.1D4, C32:1(C22:C8), C6.D6.4C4, C3:S3.2M4(2), C12.29D6:6C2, (C2xS32).2C4, (C4xS32).5C2, C3:S3.2(C2xC8), C3:S3:3C8:6C2, C2.1((S32):C4), (C4xC3:S3).50C22, (C3xC6).1(C22:C4), (C2xC3:S3).5(C2xC4), SmallGroup(288,374)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S32):C8
G = < a,b,c,d,e | a3=b2=c3=d2=e8=1, bab=a-1, ac=ca, ad=da, eae-1=c, bc=cb, bd=db, ebe-1=d, dcd=c-1, ece-1=a, ede-1=b >
Subgroups: 424 in 88 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C2 [x4], C3 [x2], C4, C4 [x2], C22 [x5], S3 [x6], C6 [x4], C8 [x2], C2xC4 [x4], C23, C32, Dic3 [x3], C12 [x3], D6 [x7], C2xC6, C2xC8 [x2], C22xC4, C3xS3 [x2], C3:S3 [x2], C3xC6, C3:C8, C24, C4xS3 [x5], C2xDic3, C2xC12, C22xS3, C22:C8, C3xDic3, C3:Dic3, C3xC12, S32 [x2], S32, S3xC6, C2xC3:S3, S3xC8, S3xC2xC4, C3xC3:C8, C32:2C8, S3xDic3, C6.D6, S3xC12, C4xC3:S3, C2xS32, C12.29D6, C3:S3:3C8, C4xS32, (S32):C8
Quotients: C1, C2 [x3], C4 [x2], C22, C8 [x2], C2xC4, D4 [x2], C22:C4, C2xC8, M4(2), C22:C8, S3wrC2, (S32):C4, (S32):C8
►On 24 points - transitive group 24T663
(2 10 22)(4 12 24)(6 14 18)(8 16 20)
(2 6)(4 8)(10 18)(12 20)(14 22)(16 24)
(1 9 21)(3 11 23)(5 13 17)(7 15 19)
(1 5)(3 7)(9 17)(11 19)(13 21)(15 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (2,10,22)(4,12,24)(6,14,18)(8,16,20), (2,6)(4,8)(10,18)(12,20)(14,22)(16,24), (1,9,21)(3,11,23)(5,13,17)(7,15,19), (1,5)(3,7)(9,17)(11,19)(13,21)(15,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (2,10,22)(4,12,24)(6,14,18)(8,16,20), (2,6)(4,8)(10,18)(12,20)(14,22)(16,24), (1,9,21)(3,11,23)(5,13,17)(7,15,19), (1,5)(3,7)(9,17)(11,19)(13,21)(15,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([(2,10,22),(4,12,24),(6,14,18),(8,16,20)], [(2,6),(4,8),(10,18),(12,20),(14,22),(16,24)], [(1,9,21),(3,11,23),(5,13,17),(7,15,19)], [(1,5),(3,7),(9,17),(11,19),(13,21),(15,23)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)])
G:=TransitiveGroup(24,663);
►On 24 points - transitive group 24T664
(2 13 22)(4 15 24)(6 9 18)(8 11 20)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)
(1 12 21)(3 14 23)(5 16 17)(7 10 19)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 23)(11 15)(12 17)(14 19)(16 21)(18 22)(20 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)
G:=sub<Sym(24)| (2,13,22)(4,15,24)(6,9,18)(8,11,20), (1,5)(2,6)(3,7)(4,8)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23), (1,12,21)(3,14,23)(5,16,17)(7,10,19), (1,5)(2,6)(3,7)(4,8)(9,13)(10,23)(11,15)(12,17)(14,19)(16,21)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (2,13,22)(4,15,24)(6,9,18)(8,11,20), (1,5)(2,6)(3,7)(4,8)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23), (1,12,21)(3,14,23)(5,16,17)(7,10,19), (1,5)(2,6)(3,7)(4,8)(9,13)(10,23)(11,15)(12,17)(14,19)(16,21)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([(2,13,22),(4,15,24),(6,9,18),(8,11,20)], [(1,5),(2,6),(3,7),(4,8),(9,22),(10,14),(11,24),(12,16),(13,18),(15,20),(17,21),(19,23)], [(1,12,21),(3,14,23),(5,16,17),(7,10,19)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,23),(11,15),(12,17),(14,19),(16,21),(18,22),(20,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)])
G:=TransitiveGroup(24,664);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 4 | 4 | 1 | 1 | 6 | 6 | 9 | 9 | 4 | 4 | 12 | 12 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | M4(2) | S3wrC2 | (S32):C4 | (S32):C4 | (S32):C8 |
| kernel | (S32):C8 | C12.29D6 | C3:S3:3C8 | C4xS32 | C6.D6 | C2xS32 | S32 | C3:Dic3 | C3xC12 | C3:S3 | C4 | C2 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 8 |
Matrix representation of (S32):C8 ►in GL4(F5) generated by
| 4 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 4 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [4,0,2,0,0,1,0,0,2,0,0,0,0,0,0,1],[0,0,3,0,0,4,0,0,2,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,3,0,0,1,0,0,3,0,4],[4,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0],[0,0,0,3,0,0,4,0,0,2,0,0,1,0,0,0] >;
(S32):C8 in GAP, Magma, Sage, TeX
(S_3^2)\rtimes C_8
% in TeX
G:=Group("(S3^2):C8"); // GroupNames label
G:=SmallGroup(288,374);
// by ID
G=gap.SmallGroup(288,374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,80,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=c,b*c=c*b,b*d=d*b,e*b*e^-1=d,d*c*d=c^-1,e*c*e^-1=a,e*d*e^-1=b>;
// generators/relations