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## G = S32⋊D4order 288 = 25·32

### The semidirect product of S32 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — S32⋊D4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — S32⋊D4
 Lower central C32 — C2×C3⋊S3 — S32⋊D4
 Upper central C1 — C2 — C4

Generators and relations for S32⋊D4
G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 872 in 148 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C4⋊D4, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×D4, S3×Dic3, C6.D6, D6⋊S3, S3×C12, C3×D12, C4×C3⋊S3, S3≀C2, C2×C32⋊C4, C2×S32, C2×S32, S32⋊C4, C4⋊(C32⋊C4), C4×S32, D6⋊D6, C2×S3≀C2, S32⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, S3≀C2, C2×S3≀C2, S32⋊D4

Character table of S32⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E size 1 1 6 6 9 9 12 12 4 4 2 6 6 18 36 36 4 4 12 12 24 24 4 4 8 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 0 0 -2 -2 0 0 2 2 -2 0 0 2 0 0 2 2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 2 -2 0 0 2 2 0 0 0 0 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 2 -2 0 0 2 2 0 0 0 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 0 0 -2 -2 0 0 2 2 2 0 0 -2 0 0 2 2 0 0 0 0 2 2 2 0 0 orthogonal lifted from D4 ρ13 2 -2 0 0 -2 2 0 0 2 2 0 -2i 2i 0 0 0 -2 -2 0 0 0 0 0 0 0 2i -2i complex lifted from C4○D4 ρ14 2 -2 0 0 -2 2 0 0 2 2 0 2i -2i 0 0 0 -2 -2 0 0 0 0 0 0 0 -2i 2i complex lifted from C4○D4 ρ15 4 4 2 2 0 0 0 0 1 -2 -4 -2 -2 0 0 0 -2 1 -1 -1 0 0 -1 -1 2 1 1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 0 0 2 -2 -2 1 -4 0 0 0 0 0 1 -2 0 0 -1 1 2 2 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 0 0 2 2 -2 1 4 0 0 0 0 0 1 -2 0 0 -1 -1 -2 -2 1 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 0 0 -2 2 -2 1 -4 0 0 0 0 0 1 -2 0 0 1 -1 2 2 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ19 4 4 0 0 0 0 -2 -2 -2 1 4 0 0 0 0 0 1 -2 0 0 1 1 -2 -2 1 0 0 orthogonal lifted from S3≀C2 ρ20 4 4 -2 -2 0 0 0 0 1 -2 -4 2 2 0 0 0 -2 1 1 1 0 0 -1 -1 2 -1 -1 orthogonal lifted from C2×S3≀C2 ρ21 4 4 2 2 0 0 0 0 1 -2 4 2 2 0 0 0 -2 1 -1 -1 0 0 1 1 -2 -1 -1 orthogonal lifted from S3≀C2 ρ22 4 4 -2 -2 0 0 0 0 1 -2 4 -2 -2 0 0 0 -2 1 1 1 0 0 1 1 -2 1 1 orthogonal lifted from S3≀C2 ρ23 4 -4 2 -2 0 0 0 0 1 -2 0 -2i 2i 0 0 0 2 -1 -1 1 0 0 3i -3i 0 -i i complex faithful ρ24 4 -4 -2 2 0 0 0 0 1 -2 0 2i -2i 0 0 0 2 -1 1 -1 0 0 3i -3i 0 i -i complex faithful ρ25 4 -4 -2 2 0 0 0 0 1 -2 0 -2i 2i 0 0 0 2 -1 1 -1 0 0 -3i 3i 0 -i i complex faithful ρ26 4 -4 2 -2 0 0 0 0 1 -2 0 2i -2i 0 0 0 2 -1 -1 1 0 0 -3i 3i 0 i -i complex faithful ρ27 8 -8 0 0 0 0 0 0 -4 2 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S32⋊D4
On 24 points - transitive group 24T644
Generators in S24
```(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 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(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 23 3 21)(2 22 4 24)(5 14 12 17)(6 13 9 20)(7 16 10 19)(8 15 11 18)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)```

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

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

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

`G:=TransitiveGroup(24,644);`

On 24 points - transitive group 24T649
Generators in S24
```(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 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(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 22)(2 21)(3 24)(4 23)(5 15 10 20)(6 14 11 19)(7 13 12 18)(8 16 9 17)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)```

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

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

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

`G:=TransitiveGroup(24,649);`

On 24 points - transitive group 24T652
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(1 24)(2 23)(3 22)(4 21)(5 13 10 18)(6 16 11 17)(7 15 12 20)(8 14 9 19)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)```

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

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

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

`G:=TransitiveGroup(24,652);`

On 24 points - transitive group 24T656
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(1 22 3 24)(2 21 4 23)(5 13 12 20)(6 16 9 19)(7 15 10 18)(8 14 11 17)
(5 12)(6 9)(7 10)(8 11)(21 23)(22 24)```

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

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

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

`G:=TransitiveGroup(24,656);`

Matrix representation of S32⋊D4 in GL4(𝔽5) generated by

 0 4 0 0 1 0 0 0 0 0 0 1 0 0 4 0
,
 3 1 1 2 4 3 2 4 1 3 0 3 3 4 2 0
,
 3 4 4 2 1 3 2 1 4 3 0 2 3 1 3 0
,
 1 0 1 2 0 4 3 1 0 0 0 2 0 0 2 0
,
 1 0 1 2 0 1 2 4 0 0 0 3 0 0 2 0
`G:=sub<GL(4,GF(5))| [0,1,0,0,4,0,0,0,0,0,0,4,0,0,1,0],[3,4,1,3,1,3,3,4,1,2,0,2,2,4,3,0],[3,1,4,3,4,3,3,1,4,2,0,3,2,1,2,0],[1,0,0,0,0,4,0,0,1,3,0,2,2,1,2,0],[1,0,0,0,0,1,0,0,1,2,0,2,2,4,3,0] >;`

S32⋊D4 in GAP, Magma, Sage, TeX

`S_3^2\rtimes D_4`
`% in TeX`

`G:=Group("S3^2:D4");`
`// GroupNames label`

`G:=SmallGroup(288,878);`
`// by ID`

`G=gap.SmallGroup(288,878);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,100,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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