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## G = S32⋊Q8order 288 = 25·32

### The semidirect product of S32 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — S32⋊Q8
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — S32⋊Q8
 Lower central C32 — C2×C3⋊S3 — S32⋊Q8
 Upper central C1 — C2 — C4

Generators and relations for S32⋊Q8
G = < a,b,c,d,e,f | a3=b2=c3=d2=e4=1, f2=e2, bab=fcf-1=a-1, ac=ca, ad=da, ae=ea, faf-1=dcd=c-1, bc=cb, bd=db, be=eb, fbf-1=d, ce=ec, de=ed, fdf-1=b, fef-1=e-1 >

Subgroups: 600 in 120 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×6], C22 [×5], S3 [×6], C6 [×4], C2×C4 [×8], Q8 [×2], C23, C32, Dic3 [×5], C12 [×5], D6 [×7], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×3], C4×S3 [×7], C2×Dic3, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3, C3×C12, C32⋊C4 [×2], S32 [×2], S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Q8, S3×Dic3, C6.D6, C6.D6 [×2], C322Q8, C3×Dic6, S3×C12, C4×C3⋊S3, C2×C32⋊C4 [×2], C2×S32, S32⋊C4 [×2], C3⋊S3.Q8 [×2], C4⋊(C32⋊C4), Dic3.D6, C4×S32, S32⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, S3≀C2, C2×S3≀C2, S32⋊Q8

Character table of S32⋊Q8

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G size 1 1 6 6 9 9 4 4 2 6 6 12 12 18 36 36 4 4 12 12 4 4 8 12 12 24 24 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 2 2 2 0 0 0 0 -2 0 0 2 2 0 0 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 -2 2 2 -2 0 0 0 0 2 0 0 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 -2 2 2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 0 0 -2 2 2 2 0 2i -2i 0 0 0 0 0 -2 -2 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ14 2 -2 0 0 -2 2 2 2 0 -2i 2i 0 0 0 0 0 -2 -2 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ15 4 4 -2 -2 0 0 -2 1 4 -2 -2 0 0 0 0 0 -2 1 1 1 1 1 -2 1 1 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 2 2 0 0 -2 1 4 2 2 0 0 0 0 0 -2 1 -1 -1 1 1 -2 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ17 4 4 0 0 0 0 1 -2 -4 0 0 -2 2 0 0 0 1 -2 0 0 2 2 -1 0 0 -1 1 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 0 0 0 1 -2 4 0 0 2 2 0 0 0 1 -2 0 0 -2 -2 1 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ19 4 4 0 0 0 0 1 -2 4 0 0 -2 -2 0 0 0 1 -2 0 0 -2 -2 1 0 0 1 1 orthogonal lifted from S3≀C2 ρ20 4 4 2 2 0 0 -2 1 -4 -2 -2 0 0 0 0 0 -2 1 -1 -1 -1 -1 2 1 1 0 0 orthogonal lifted from C2×S3≀C2 ρ21 4 4 -2 -2 0 0 -2 1 -4 2 2 0 0 0 0 0 -2 1 1 1 -1 -1 2 -1 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ22 4 4 0 0 0 0 1 -2 -4 0 0 2 -2 0 0 0 1 -2 0 0 2 2 -1 0 0 1 -1 orthogonal lifted from C2×S3≀C2 ρ23 4 -4 -2 2 0 0 -2 1 0 2i -2i 0 0 0 0 0 2 -1 -1 1 -3i 3i 0 -i i 0 0 complex faithful ρ24 4 -4 2 -2 0 0 -2 1 0 -2i 2i 0 0 0 0 0 2 -1 1 -1 -3i 3i 0 i -i 0 0 complex faithful ρ25 4 -4 -2 2 0 0 -2 1 0 -2i 2i 0 0 0 0 0 2 -1 -1 1 3i -3i 0 i -i 0 0 complex faithful ρ26 4 -4 2 -2 0 0 -2 1 0 2i -2i 0 0 0 0 0 2 -1 1 -1 3i -3i 0 -i i 0 0 complex faithful ρ27 8 -8 0 0 0 0 2 -4 0 0 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of S32⋊Q8
On 24 points - transitive group 24T641
Generators in S24
```(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 20)(14 17)(15 18)(16 19)(21 23)(22 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)```

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

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

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

`G:=TransitiveGroup(24,641);`

On 24 points - transitive group 24T648
Generators in S24
```(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)(17 19)(18 20)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(5 7)(6 8)(9 11)(10 12)(13 18)(14 19)(15 20)(16 17)(21 23)(22 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)```

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

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

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

`G:=TransitiveGroup(24,648);`

Matrix representation of S32⋊Q8 in GL4(𝔽5) generated by

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

S32⋊Q8 in GAP, Magma, Sage, TeX

`S_3^2\rtimes Q_8`
`% in TeX`

`G:=Group("S3^2:Q8");`
`// GroupNames label`

`G:=SmallGroup(288,868);`
`// by ID`

`G=gap.SmallGroup(288,868);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,219,100,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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