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G = S32×D4  order 288 = 25·32

Direct product of S3, S3 and D4

direct product, metabelian, supersoluble, monomial, rational

Aliases: S32×D4, C62⋊C23, D12⋊10D6, (C3×D4)⋊7D6, (C4×S3)⋊6D6, C3⋊D4⋊1D6, (C3×C12)⋊C23, (S3×D12)⋊9C2, Dic3⋊D6⋊4C2, (S3×C6)⋊3C23, C12⋊1(C22×S3), D6⋊3(C22×S3), D6⋊D6⋊11C2, C32⋊4(C22×D4), (S3×C12)⋊4C22, (C22×S3)⋊11D6, (C3×C6).17C24, C6.17(S3×C23), C3⋊Dic3⋊1C23, C12⋊S3⋊9C22, (C3×D12)⋊12C22, D6⋊S3⋊4C22, C3⋊D12⋊3C22, Dic3⋊1(C22×S3), (S3×Dic3)⋊7C22, (C3×Dic3)⋊1C23, C6.D6⋊7C22, (D4×C32)⋊9C22, C32⋊7D4⋊1C22, C4⋊1(C2×S32), C3⋊3(C2×S3×D4), (C4×S32)⋊4C2, (C3×S3×D4)⋊7C2, C3⋊S3⋊3(C2×D4), (D4×C3⋊S3)⋊6C2, C22⋊2(C2×S32), (C22×S32)⋊7C2, (C3×S3)⋊2(C2×D4), (S3×C3⋊D4)⋊2C2, (S3×C2×C6)⋊9C22, (C2×S32)⋊10C22, (C2×C3⋊S3)⋊2C23, (C4×C3⋊S3)⋊2C22, (C2×C6)⋊4(C22×S3), C2.19(C22×S32), (C3×C3⋊D4)⋊1C22, (C22×C3⋊S3)⋊7C22, SmallGroup(288,958)

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, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22×D4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C2×D12, S3×D4, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C2×S32, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×S3×D4, C4×S32, S3×D12, D6⋊D6, S3×C3⋊D4, Dic3⋊D6, C3×S3×D4, D4×C3⋊S3, C22×S32, S32×D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S32, S3×D4, S3×C23, C2×S32, C2×S3×D4, 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 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B3C4A4B4C4D6A6B6C···6G6H6I6J6K6L6M6N6O6P6Q12A12B12C12D12E
order12222222222222223334444666···666666666661212121212
size11223333666699181822426618224···4666688121212124481212

45 irreducible representations

dim111111111222222244448
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D6D6D6D6D6S32S3×D4C2×S32C2×S32S32×D4
kernelS32×D4C4×S32S3×D12D6⋊D6S3×C3⋊D4Dic3⋊D6C3×S3×D4D4×C3⋊S3C22×S32S3×D4S32C4×S3D12C3⋊D4C3×D4C22×S3D4S3C4C22C1
# reps112142212242242414121

Matrix representation of S32×D4 ►in GL6(ℤ)

100000
010000
001000
000100
000001
0000-1-1
,
-100000
0-10000
00-1000
000-100
000010
0000-1-1
,
-110000
-100000
001000
000100
000010
000001
,
0-10000
-100000
00-1000
000-100
000010
000001
,
100000
010000
000-100
001000
000010
000001
,
100000
010000
000100
001000
000010
000001

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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