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## G = C3⋊S3.4D8order 288 = 25·32

### The non-split extension by C3⋊S3 of D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3⋊S3.4D8
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4⋊(C32⋊C4) — C3⋊S3.4D8
 Lower central C32 — C3×C6 — C3×C12 — C3⋊S3.4D8
 Upper central C1 — C2 — C4 — C8

Generators and relations for C3⋊S3.4D8
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=a-1, ad=da, eae-1=ab-1, cbc=b-1, bd=db, ebe-1=a-1b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 328 in 58 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×3], C22, S3 [×4], C6 [×2], C8, C8, C2×C4 [×3], C32, Dic3 [×2], C12 [×2], D6 [×2], C4⋊C4 [×2], C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×2], C2.D8, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, S3×C8 [×2], C324C8, C3×C24, C4×C3⋊S3, C2×C32⋊C4 [×2], C8×C3⋊S3, C4⋊(C32⋊C4) [×2], C3⋊S3.4D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, D8, Q16, C2.D8, C32⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), C3⋊S3.4D8

Character table of C3⋊S3.4D8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 9 9 4 4 2 18 36 36 36 36 4 4 2 2 18 18 4 4 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 -1 i -i -i i 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 1 1 -1 -i -i i i 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 1 1 -1 i i -i -i 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 1 -1 -i i i -i 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 2 -2 -2 0 0 0 0 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -√2 √2 -√2 √2 0 0 0 0 √2 √2 -√2 -√2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ11 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 √2 -√2 √2 -√2 0 0 0 0 -√2 -√2 √2 √2 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 √2 -√2 -√2 √2 0 0 0 0 -√2 -√2 √2 √2 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 -√2 √2 √2 -√2 0 0 0 0 √2 √2 -√2 -√2 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 2 2 -2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 4 4 0 0 -2 1 4 0 0 0 0 0 1 -2 4 4 0 0 1 -2 1 -2 -2 1 -2 -2 1 1 1 -2 orthogonal lifted from C32⋊C4 ρ16 4 4 0 0 -2 1 4 0 0 0 0 0 1 -2 -4 -4 0 0 1 -2 1 -2 2 -1 2 2 -1 -1 -1 2 orthogonal lifted from C2×C32⋊C4 ρ17 4 4 0 0 1 -2 4 0 0 0 0 0 -2 1 4 4 0 0 -2 1 -2 1 1 -2 1 1 -2 -2 -2 1 orthogonal lifted from C32⋊C4 ρ18 4 4 0 0 1 -2 4 0 0 0 0 0 -2 1 -4 -4 0 0 -2 1 -2 1 -1 2 -1 -1 2 2 2 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 0 1 -2 -4 0 0 0 0 0 -2 1 0 0 0 0 2 -1 2 -1 -3i 0 3i -3i 0 0 0 3i complex lifted from C4⋊(C32⋊C4) ρ20 4 4 0 0 -2 1 -4 0 0 0 0 0 1 -2 0 0 0 0 -1 2 -1 2 0 3i 0 0 3i -3i -3i 0 complex lifted from C4⋊(C32⋊C4) ρ21 4 4 0 0 -2 1 -4 0 0 0 0 0 1 -2 0 0 0 0 -1 2 -1 2 0 -3i 0 0 -3i 3i 3i 0 complex lifted from C4⋊(C32⋊C4) ρ22 4 4 0 0 1 -2 -4 0 0 0 0 0 -2 1 0 0 0 0 2 -1 2 -1 3i 0 -3i 3i 0 0 0 -3i complex lifted from C4⋊(C32⋊C4) ρ23 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 2√2 -2√2 0 0 0 -3i 0 3i ζ87+2ζ85 √2 2ζ87+ζ85 ζ83+2ζ8 -√2 -√2 √2 2ζ83+ζ8 complex faithful ρ24 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 2√2 -2√2 0 0 3i 0 -3i 0 √2 ζ87+2ζ85 -√2 -√2 ζ83+2ζ8 2ζ87+ζ85 2ζ83+ζ8 √2 complex faithful ρ25 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 -2√2 2√2 0 0 0 3i 0 -3i 2ζ87+ζ85 -√2 ζ87+2ζ85 2ζ83+ζ8 √2 √2 -√2 ζ83+2ζ8 complex faithful ρ26 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 2√2 -2√2 0 0 -3i 0 3i 0 √2 2ζ83+ζ8 -√2 -√2 2ζ87+ζ85 ζ83+2ζ8 ζ87+2ζ85 √2 complex faithful ρ27 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 -2√2 2√2 0 0 0 -3i 0 3i ζ83+2ζ8 -√2 2ζ83+ζ8 ζ87+2ζ85 √2 √2 -√2 2ζ87+ζ85 complex faithful ρ28 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 -2√2 2√2 0 0 3i 0 -3i 0 -√2 ζ83+2ζ8 √2 √2 ζ87+2ζ85 2ζ83+ζ8 2ζ87+ζ85 -√2 complex faithful ρ29 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 -2√2 2√2 0 0 -3i 0 3i 0 -√2 2ζ87+ζ85 √2 √2 2ζ83+ζ8 ζ87+2ζ85 ζ83+2ζ8 -√2 complex faithful ρ30 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 2√2 -2√2 0 0 0 3i 0 -3i 2ζ83+ζ8 √2 ζ83+2ζ8 2ζ87+ζ85 -√2 -√2 √2 ζ87+2ζ85 complex faithful

Smallest permutation representation of C3⋊S3.4D8
On 48 points
Generators in S48
```(1 25 38)(2 26 39)(3 27 40)(4 28 33)(5 29 34)(6 30 35)(7 31 36)(8 32 37)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 31 43 36)(18 30 44 35)(19 29 45 34)(20 28 46 33)(21 27 47 40)(22 26 48 39)(23 25 41 38)(24 32 42 37)```

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

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

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

Matrix representation of C3⋊S3.4D8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 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 72 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72
,
 0 41 0 0 0 0 16 41 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 0 0 27
,
 37 40 0 0 0 0 57 36 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 46 27 0 0 0 0 0 27 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[37,57,0,0,0,0,40,36,0,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27,0,0,27,0,0,0,0,0,0,27,0,0] >;`

C3⋊S3.4D8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3._4D_8`
`% in TeX`

`G:=Group("C3:S3.4D8");`
`// GroupNames label`

`G:=SmallGroup(288,417);`
`// by ID`

`G=gap.SmallGroup(288,417);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,675,80,9413,691,12550,2372]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=c,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e^-1=a*b^-1,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^-1*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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