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## G = S32×D4order 288 = 25·32

### Direct product of S3, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S32×D4
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — S32×D4
 Lower central C32 — C3×C6 — S32×D4
 Upper central C1 — C2 — D4

Generators and relations for S32×D4
G = < a,b,c,d,e,f | a3=b2=c3=d2=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2178 in 499 conjugacy classes, 122 normal (18 characteristic)
C1, C2, C2 [×14], C3 [×2], C3, C4, C4 [×3], C22 [×2], C22 [×37], S3 [×4], S3 [×16], C6 [×2], C6 [×15], C2×C4 [×6], D4, D4 [×15], C23 [×21], C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×6], D6 [×61], C2×C6 [×4], C2×C6 [×16], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3×S3 [×4], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×2], C4×S3 [×7], D12 [×2], D12 [×7], C2×Dic3 [×2], C3⋊D4 [×4], C3⋊D4 [×14], C2×C12 [×2], C3×D4 [×2], C3×D4 [×7], C22×S3 [×4], C22×S3 [×36], C22×C6 [×4], C22×D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×4], S32 [×12], S3×C6 [×6], S3×C6 [×8], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62 [×2], S3×C2×C4 [×2], C2×D12 [×2], S3×D4 [×2], S3×D4 [×15], C2×C3⋊D4 [×4], C6×D4 [×2], S3×C23 [×4], S3×Dic3 [×2], C6.D6, D6⋊S3 [×2], C3⋊D12 [×4], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C2×S32, C2×S32 [×6], C2×S32 [×8], S3×C2×C6 [×4], C22×C3⋊S3 [×2], C2×S3×D4 [×2], C4×S32, S3×D12 [×2], D6⋊D6, S3×C3⋊D4 [×4], Dic3⋊D6 [×2], C3×S3×D4 [×2], D4×C3⋊S3, C22×S32 [×2], S32×D4
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C22×S3 [×14], C22×D4, S32, S3×D4 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×D4 [×2], C22×S32, S32×D4

Permutation representations of S32×D4
On 24 points - transitive group 24T606
Generators in S24
(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)
(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 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,606);

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 12A 12B 12C 12D 12E order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 size 1 1 2 2 3 3 3 3 6 6 6 6 9 9 18 18 2 2 4 2 6 6 18 2 2 4 ··· 4 6 6 6 6 8 8 12 12 12 12 4 4 8 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 D6 S32 S3×D4 C2×S32 C2×S32 S32×D4 kernel S32×D4 C4×S32 S3×D12 D6⋊D6 S3×C3⋊D4 Dic3⋊D6 C3×S3×D4 D4×C3⋊S3 C22×S32 S3×D4 S32 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 D4 S3 C4 C22 C1 # reps 1 1 2 1 4 2 2 1 2 2 4 2 2 4 2 4 1 4 1 2 1

Matrix representation of S32×D4 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1
,
 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S32×D4 in GAP, Magma, Sage, TeX

S_3^2\times D_4
% in TeX

G:=Group("S3^2xD4");
// GroupNames label

G:=SmallGroup(288,958);
// by ID

G=gap.SmallGroup(288,958);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,185,1356,9414]);
// Polycyclic

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

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