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## G = C32⋊C4⋊Q8order 288 = 25·32

### 1st semidirect product of C32⋊C4 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C32⋊C4⋊Q8
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C6.D6 — C3⋊S3.Q8 — C32⋊C4⋊Q8
 Lower central C32 — C2×C3⋊S3 — C32⋊C4⋊Q8
 Upper central C1 — C2 — C4

Generators and relations for C32⋊C4⋊Q8
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=d2, cbc-1=ab=ba, cac-1=a-1b, ad=da, ae=ea, bd=db, ebe-1=a-1b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 504 in 106 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×9], C22, S3 [×4], C6 [×2], C2×C4 [×7], Q8 [×4], C32, Dic3 [×6], C12 [×6], D6 [×2], C42, C4⋊C4 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], C4⋊Q8, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C32⋊C4 [×4], C2×C3⋊S3, S3×Q8 [×2], C6.D6 [×4], C322Q8 [×2], C3×Dic6 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C3⋊S3.Q8 [×4], C4×C32⋊C4, Dic3.D6 [×2], C32⋊C4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C2×D4, C2×Q8 [×2], C4⋊Q8, S3≀C2, C2×S3≀C2, C32⋊C4⋊Q8

Character table of C32⋊C4⋊Q8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 12A 12B 12C 12D 12E 12F size 1 1 9 9 4 4 2 12 12 12 12 18 18 18 18 18 4 4 8 8 24 24 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 -2 2 2 2 0 0 0 0 -2 0 0 0 0 2 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 2 -2 0 0 0 0 2 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 2 0 0 0 0 0 0 2 -2 0 0 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 2 0 0 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 -2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 4 4 0 0 -2 1 -4 0 0 -2 2 0 0 0 0 0 1 -2 -1 2 1 0 -1 0 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 -2 1 4 0 0 2 2 0 0 0 0 0 1 -2 1 -2 -1 0 -1 0 orthogonal lifted from S3≀C2 ρ17 4 4 0 0 1 -2 4 2 2 0 0 0 0 0 0 0 -2 1 -2 1 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 1 -2 -4 -2 2 0 0 0 0 0 0 0 -2 1 2 -1 0 1 0 -1 orthogonal lifted from C2×S3≀C2 ρ19 4 4 0 0 -2 1 -4 0 0 2 -2 0 0 0 0 0 1 -2 -1 2 -1 0 1 0 orthogonal lifted from C2×S3≀C2 ρ20 4 4 0 0 -2 1 4 0 0 -2 -2 0 0 0 0 0 1 -2 1 -2 1 0 1 0 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 1 -2 -4 2 -2 0 0 0 0 0 0 0 -2 1 2 -1 0 -1 0 1 orthogonal lifted from C2×S3≀C2 ρ22 4 4 0 0 1 -2 4 -2 -2 0 0 0 0 0 0 0 -2 1 -2 1 0 1 0 1 orthogonal lifted from S3≀C2 ρ23 8 -8 0 0 -4 2 0 0 0 0 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 8 -8 0 0 2 -4 0 0 0 0 0 0 0 0 0 0 4 -2 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊C4⋊Q8
On 48 points
Generators in S48
```(1 24 22)(2 23 21)(3 42 44)(4 41 43)(5 37 39)(6 40 38)(7 45 47)(8 46 48)(9 19 17)(10 18 20)(11 26 28)(12 25 27)(13 29 31)(14 32 30)(15 33 35)(16 34 36)
(2 21 23)(4 43 41)(6 38 40)(7 47 45)(10 20 18)(12 27 25)(14 30 32)(15 35 33)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 9 8 5)(2 10 7 6)(3 11 16 13)(4 12 15 14)(17 48 39 22)(18 45 40 23)(19 46 37 24)(20 47 38 21)(25 33 32 41)(26 34 29 42)(27 35 30 43)(28 36 31 44)
(1 15 8 4)(2 16 7 3)(5 14 9 12)(6 13 10 11)(17 27 39 30)(18 26 40 29)(19 25 37 32)(20 28 38 31)(21 36 47 44)(22 35 48 43)(23 34 45 42)(24 33 46 41)```

`G:=sub<Sym(48)| (1,24,22)(2,23,21)(3,42,44)(4,41,43)(5,37,39)(6,40,38)(7,45,47)(8,46,48)(9,19,17)(10,18,20)(11,26,28)(12,25,27)(13,29,31)(14,32,30)(15,33,35)(16,34,36), (2,21,23)(4,43,41)(6,38,40)(7,47,45)(10,20,18)(12,27,25)(14,30,32)(15,35,33), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,9,8,5)(2,10,7,6)(3,11,16,13)(4,12,15,14)(17,48,39,22)(18,45,40,23)(19,46,37,24)(20,47,38,21)(25,33,32,41)(26,34,29,42)(27,35,30,43)(28,36,31,44), (1,15,8,4)(2,16,7,3)(5,14,9,12)(6,13,10,11)(17,27,39,30)(18,26,40,29)(19,25,37,32)(20,28,38,31)(21,36,47,44)(22,35,48,43)(23,34,45,42)(24,33,46,41)>;`

`G:=Group( (1,24,22)(2,23,21)(3,42,44)(4,41,43)(5,37,39)(6,40,38)(7,45,47)(8,46,48)(9,19,17)(10,18,20)(11,26,28)(12,25,27)(13,29,31)(14,32,30)(15,33,35)(16,34,36), (2,21,23)(4,43,41)(6,38,40)(7,47,45)(10,20,18)(12,27,25)(14,30,32)(15,35,33), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,9,8,5)(2,10,7,6)(3,11,16,13)(4,12,15,14)(17,48,39,22)(18,45,40,23)(19,46,37,24)(20,47,38,21)(25,33,32,41)(26,34,29,42)(27,35,30,43)(28,36,31,44), (1,15,8,4)(2,16,7,3)(5,14,9,12)(6,13,10,11)(17,27,39,30)(18,26,40,29)(19,25,37,32)(20,28,38,31)(21,36,47,44)(22,35,48,43)(23,34,45,42)(24,33,46,41) );`

`G=PermutationGroup([(1,24,22),(2,23,21),(3,42,44),(4,41,43),(5,37,39),(6,40,38),(7,45,47),(8,46,48),(9,19,17),(10,18,20),(11,26,28),(12,25,27),(13,29,31),(14,32,30),(15,33,35),(16,34,36)], [(2,21,23),(4,43,41),(6,38,40),(7,47,45),(10,20,18),(12,27,25),(14,30,32),(15,35,33)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,9,8,5),(2,10,7,6),(3,11,16,13),(4,12,15,14),(17,48,39,22),(18,45,40,23),(19,46,37,24),(20,47,38,21),(25,33,32,41),(26,34,29,42),(27,35,30,43),(28,36,31,44)], [(1,15,8,4),(2,16,7,3),(5,14,9,12),(6,13,10,11),(17,27,39,30),(18,26,40,29),(19,25,37,32),(20,28,38,31),(21,36,47,44),(22,35,48,43),(23,34,45,42),(24,33,46,41)])`

Matrix representation of C32⋊C4⋊Q8 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 12 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
,
 9 3 0 0 0 0 3 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,12,0,0,0,0,0,12,0,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C32⋊C4⋊Q8 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_4\rtimes Q_8`
`% in TeX`

`G:=Group("C3^2:C4:Q8");`
`// GroupNames label`

`G:=SmallGroup(288,870);`
`// by ID`

`G=gap.SmallGroup(288,870);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,422,219,100,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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