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## G = C12⋊S3.C4order 288 = 25·32

### The non-split extension by C12⋊S3 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12⋊S3.C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C3⋊S3⋊3C8 — C12⋊S3.C4
 Lower central C32 — C3×C6 — C12⋊S3.C4
 Upper central C1 — C2 — Q8

Generators and relations for C12⋊S3.C4
G = < a,b,c,d | a12=b3=c2=1, d4=a6, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a8b-1, cd=dc >

Subgroups: 520 in 102 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C8, M4(2), C4○D4, C3⋊S3, C3×C6, C4×S3, D12, C3×Q8, C8○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, Q83S3, C322C8, C322C8, C4×C3⋊S3, C12⋊S3, Q8×C32, C3⋊S33C8, C32⋊M4(2), C12.26D6, C12⋊S3.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, C12⋊S3.C4

Character table of C12⋊S3.C4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E 12F size 1 1 18 18 18 4 4 2 2 2 9 9 4 4 9 9 9 9 18 18 18 18 18 18 8 8 8 8 8 8 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 -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 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 -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 1 1 -1 linear of order 2 ρ9 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -i -i i i i i -i i -i -i 1 1 1 1 1 1 linear of order 4 ρ10 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 i i -i -i -i -i i -i i i 1 1 1 1 1 1 linear of order 4 ρ11 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 i i -i -i i i i -i -i -i -1 1 1 -1 -1 -1 linear of order 4 ρ12 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 -i -i i i -i -i -i i i i -1 1 1 -1 -1 -1 linear of order 4 ρ13 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 i i -i -i -i i -i i i -i 1 -1 -1 -1 -1 1 linear of order 4 ρ14 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -i -i i i i -i i -i -i i 1 -1 -1 -1 -1 1 linear of order 4 ρ15 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -i -i i i -i i i -i i -i -1 -1 -1 1 1 -1 linear of order 4 ρ16 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 i i -i -i i -i -i i -i i -1 -1 -1 1 1 -1 linear of order 4 ρ17 2 -2 0 0 0 2 2 0 0 0 -2i 2i -2 -2 2ζ83 2ζ87 2ζ8 2ζ85 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ18 2 -2 0 0 0 2 2 0 0 0 2i -2i -2 -2 2ζ8 2ζ85 2ζ83 2ζ87 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ19 2 -2 0 0 0 2 2 0 0 0 -2i 2i -2 -2 2ζ87 2ζ83 2ζ85 2ζ8 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ20 2 -2 0 0 0 2 2 0 0 0 2i -2i -2 -2 2ζ85 2ζ8 2ζ87 2ζ83 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ21 4 4 0 0 0 1 -2 4 -4 -4 0 0 -2 1 0 0 0 0 0 0 0 0 0 0 2 -1 2 1 -2 -1 orthogonal lifted from C2×C32⋊C4 ρ22 4 4 0 0 0 1 -2 -4 4 -4 0 0 -2 1 0 0 0 0 0 0 0 0 0 0 2 1 -2 -1 2 -1 orthogonal lifted from C2×C32⋊C4 ρ23 4 4 0 0 0 -2 1 4 4 4 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 1 -2 1 -2 1 -2 orthogonal lifted from C32⋊C4 ρ24 4 4 0 0 0 -2 1 -4 -4 4 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 1 2 -1 2 -1 -2 orthogonal lifted from C2×C32⋊C4 ρ25 4 4 0 0 0 -2 1 -4 4 -4 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 -1 -2 1 2 -1 2 orthogonal lifted from C2×C32⋊C4 ρ26 4 4 0 0 0 -2 1 4 -4 -4 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 -1 2 -1 -2 1 2 orthogonal lifted from C2×C32⋊C4 ρ27 4 4 0 0 0 1 -2 -4 -4 4 0 0 -2 1 0 0 0 0 0 0 0 0 0 0 -2 -1 2 -1 2 1 orthogonal lifted from C2×C32⋊C4 ρ28 4 4 0 0 0 1 -2 4 4 4 0 0 -2 1 0 0 0 0 0 0 0 0 0 0 -2 1 -2 1 -2 1 orthogonal lifted from C32⋊C4 ρ29 8 -8 0 0 0 -4 2 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ30 8 -8 0 0 0 2 -4 0 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of C12⋊S3.C4
On 48 points
Generators in S48
```(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 20 9)(2 17 10)(3 18 11)(4 19 12)(5 16 22)(6 13 23)(7 14 24)(8 15 21)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(2 4)(5 14)(6 13)(7 16)(8 15)(9 20)(10 19)(11 18)(12 17)(22 24)(25 33)(26 32)(27 31)(28 30)(34 36)(37 43)(38 42)(39 41)(44 48)(45 47)
(1 35 23 46 3 29 21 40)(2 32 24 43 4 26 22 37)(5 45 10 36 7 39 12 30)(6 42 11 33 8 48 9 27)(13 38 18 25 15 44 20 31)(14 47 19 34 16 41 17 28)```

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

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

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

Matrix representation of C12⋊S3.C4 in GL6(𝔽73)

 51 71 0 0 0 0 60 22 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 46 46 1 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 27 0 72 72 0 0 0 46 1 0
,
 1 0 0 0 0 0 51 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 27 27 72 72
,
 22 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 72 1 0 0 27 27 71 72 0 0 48 48 46 0 0 0 48 49 46 0

`G:=sub<GL(6,GF(73))| [51,60,0,0,0,0,71,22,0,0,0,0,0,0,72,0,46,0,0,0,0,72,46,0,0,0,0,0,1,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,27,0,0,0,72,72,0,46,0,0,0,0,72,1,0,0,0,0,72,0],[1,51,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,27,0,0,1,0,0,27,0,0,0,0,1,72,0,0,0,0,0,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,27,48,48,0,0,0,27,48,49,0,0,72,71,46,46,0,0,1,72,0,0] >;`

C12⋊S3.C4 in GAP, Magma, Sage, TeX

`C_{12}\rtimes S_3.C_4`
`% in TeX`

`G:=Group("C12:S3.C4");`
`// GroupNames label`

`G:=SmallGroup(288,937);`
`// by ID`

`G=gap.SmallGroup(288,937);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,219,100,80,9413,362,12550,1203]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^3=c^2=1,d^4=a^6,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^8*b^-1,c*d=d*c>;`
`// generators/relations`

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