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## G = C12⋊3D4order 96 = 25·3

### 3rd semidirect product of C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — C12⋊3D4
 Lower central C3 — C2×C6 — C12⋊3D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C123D4
G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 290 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×D4, C2×D4, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C41D4, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, C123D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C41D4, S3×D4, C2×C3⋊D4, C123D4

Character table of C123D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 4 4 12 12 2 2 2 6 6 6 6 2 2 2 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 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 0 0 0 0 2 0 0 0 2 0 -2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 2 0 0 -2 0 2 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 2 0 0 0 -2 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 0 0 0 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 0 0 0 2 -2 2 0 0 0 0 -2 -2 2 0 0 0 0 2 -2 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 -2 -2 2 0 0 0 0 2 0 0 2 0 -2 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 -2 -2 0 0 -1 2 2 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from D6 ρ19 2 2 -2 -2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 -√-3 -√-3 √-3 √-3 1 -1 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 √-3 √-3 -√-3 -√-3 1 -1 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 0 0 -1 -2 2 0 0 0 0 1 1 -1 √-3 -√-3 √-3 -√-3 -1 1 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 0 0 0 0 -1 -2 2 0 0 0 0 1 1 -1 -√-3 √-3 -√-3 √-3 -1 1 complex lifted from C3⋊D4 ρ23 4 -4 -4 4 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4

Smallest permutation representation of C123D4
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 47 34 24)(2 40 35 17)(3 45 36 22)(4 38 25 15)(5 43 26 20)(6 48 27 13)(7 41 28 18)(8 46 29 23)(9 39 30 16)(10 44 31 21)(11 37 32 14)(12 42 33 19)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 31)(26 30)(27 29)(32 36)(33 35)```

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

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

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

Matrix representation of C123D4 in GL4(𝔽13) generated by

 0 12 0 0 1 0 0 0 0 0 12 1 0 0 12 0
,
 0 12 0 0 1 0 0 0 0 0 4 11 0 0 2 9
,
 1 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,0,0,0,0,0,4,2,0,0,11,9],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;`

C123D4 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_3D_4`
`% in TeX`

`G:=Group("C12:3D4");`
`// GroupNames label`

`G:=SmallGroup(96,147);`
`// by ID`

`G=gap.SmallGroup(96,147);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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