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G = S32⋊C8order 288 = 25·32

The semidirect product of S32 and C8 acting via C8/C4=C2

non-abelian, soluble, monomial

Aliases: S32⋊C8, C4.16S3≀C2, (C3×C12).16D4, C3⋊Dic3.1D4, C321(C22⋊C8), C6.D6.4C4, C3⋊S3.2M4(2), C12.29D66C2, (C4×S32).5C2, (C2×S32).2C4, C2.1(S32⋊C4), C3⋊S3.2(C2×C8), C3⋊S33C86C2, (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

C1C32C3⋊S3 — S32⋊C8
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4×S32 — S32⋊C8
C32C3⋊S3 — S32⋊C8
C1C4

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 [×4], C3 [×2], C4, C4 [×2], C22 [×5], S3 [×6], C6 [×4], C8 [×2], C2×C4 [×4], C23, C32, Dic3 [×3], C12 [×3], D6 [×7], C2×C6, C2×C8 [×2], C22×C4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24, C4×S3 [×5], C2×Dic3, C2×C12, C22×S3, C22⋊C8, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S32, S3×C6, C2×C3⋊S3, S3×C8, S3×C2×C4, C3×C3⋊C8, C322C8, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C12.29D6, C3⋊S33C8, C4×S32, S32⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), C22⋊C8, S3≀C2, S32⋊C4, S32⋊C8

Permutation representations of S32⋊C8
On 24 points - transitive group 24T663
Generators in S24
(2 14 22)(4 16 24)(6 10 18)(8 12 20)
(2 6)(4 8)(10 22)(12 24)(14 18)(16 20)
(1 13 21)(3 15 23)(5 9 17)(7 11 19)
(1 5)(3 7)(9 21)(11 23)(13 17)(15 19)
(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), (2,6)(4,8)(10,22)(12,24)(14,18)(16,20), (1,13,21)(3,15,23)(5,9,17)(7,11,19), (1,5)(3,7)(9,21)(11,23)(13,17)(15,19), (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), (2,6)(4,8)(10,22)(12,24)(14,18)(16,20), (1,13,21)(3,15,23)(5,9,17)(7,11,19), (1,5)(3,7)(9,21)(11,23)(13,17)(15,19), (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)], [(2,6),(4,8),(10,22),(12,24),(14,18),(16,20)], [(1,13,21),(3,15,23),(5,9,17),(7,11,19)], [(1,5),(3,7),(9,21),(11,23),(13,17),(15,19)], [(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
Generators in S24
(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 2A2B2C2D2E3A3B4A4B4C4D4E4F6A6B6C6D8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A24B24C24D
order1222223344444466668888888812121212121224242424
size116699441166994412126666181818184444121212121212

36 irreducible representations

dim11111112224444
type++++++++
imageC1C2C2C2C4C4C8D4D4M4(2)S3≀C2S32⋊C4S32⋊C4S32⋊C8
kernelS32⋊C8C12.29D6C3⋊S33C8C4×S32C6.D6C2×S32S32C3⋊Dic3C3×C12C3⋊S3C4C2C2C1
# reps11112281124228

Matrix representation of S32⋊C8 in GL4(𝔽5) generated by

4020
0100
2000
0001
,
0020
0400
3000
0004
,
1000
0003
0010
0304
,
4000
0002
0040
0300
,
0001
0020
0400
3000
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

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