non-abelian, soluble, monomial
Aliases: S32⋊C8, C4.16S3≀C2, (C3×C12).16D4, C3⋊Dic3.1D4, C32⋊1(C22⋊C8), C6.D6.4C4, C3⋊S3.2M4(2), C12.29D6⋊6C2, (C4×S32).5C2, (C2×S32).2C4, C2.1(S32⋊C4), C3⋊S3.2(C2×C8), C3⋊S3⋊3C8⋊6C2, (C4×C3⋊S3).50C22, (C3×C6).1(C22⋊C4), (C2×C3⋊S3).5(C2×C4), 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, C3, C4, C4, C22, S3, C6, C8, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C22×C4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊C8, C3×Dic3, C3⋊Dic3, C3×C12, S32, S32, S3×C6, C2×C3⋊S3, S3×C8, S3×C2×C4, C3×C3⋊C8, C32⋊2C8, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C12.29D6, C3⋊S3⋊3C8, C4×S32, S32⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, S3≀C2, 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 14 22)(4 16 24)(6 10 18)(8 12 20)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)
(1 13 21)(3 15 23)(5 9 17)(7 11 19)
(1 5)(2 6)(3 7)(4 8)(9 21)(10 14)(11 23)(12 16)(13 17)(15 19)(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)| (2,14,22)(4,16,24)(6,10,18)(8,12,20), (1,5)(2,6)(3,7)(4,8)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (1,13,21)(3,15,23)(5,9,17)(7,11,19), (1,5)(2,6)(3,7)(4,8)(9,21)(10,14)(11,23)(12,16)(13,17)(15,19)(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( (2,14,22)(4,16,24)(6,10,18)(8,12,20), (1,5)(2,6)(3,7)(4,8)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (1,13,21)(3,15,23)(5,9,17)(7,11,19), (1,5)(2,6)(3,7)(4,8)(9,21)(10,14)(11,23)(12,16)(13,17)(15,19)(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([[(2,14,22),(4,16,24),(6,10,18),(8,12,20)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,22),(11,15),(12,24),(14,18),(16,20),(17,21),(19,23)], [(1,13,21),(3,15,23),(5,9,17),(7,11,19)], [(1,5),(2,6),(3,7),(4,8),(9,21),(10,14),(11,23),(12,16),(13,17),(15,19),(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) | S3≀C2 | S32⋊C4 | S32⋊C4 | S32⋊C8 |
| kernel | S32⋊C8 | C12.29D6 | C3⋊S3⋊3C8 | C4×S32 | C6.D6 | C2×S32 | S32 | C3⋊Dic3 | C3×C12 | 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