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## G = C4.D12order 96 = 25·3

### 5th non-split extension by C4 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4.D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C4.D12
 Lower central C3 — C2×C6 — C4.D12
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4.D12
G = < a,b,c | a4=b12=1, c2=a2, bab-1=cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 170 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊Q8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C4.D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×D12, D42S3, S3×Q8, C4.D12

Character table of C4.D12

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 6 6 2 2 2 4 4 6 6 12 12 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 -1 -2 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 -2 2 -2 -2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -1 2 2 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 -2 2 -2 2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 -1 √3 √3 1 -√3 -√3 orthogonal lifted from D12 ρ15 2 2 2 2 0 0 -1 -2 -2 2 -2 0 0 0 0 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from D6 ρ16 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 1 √3 -√3 -1 √3 -√3 orthogonal lifted from D12 ρ17 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 1 -√3 √3 -1 -√3 √3 orthogonal lifted from D12 ρ18 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 -1 -√3 -√3 1 √3 √3 orthogonal lifted from D12 ρ19 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 2 -2 -2 2 0 0 2 0 0 0 0 2i -2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 2 0 0 0 0 -2i 2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 -4 4 0 0 -2 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ24 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of C4.D12
On 48 points
Generators in S48
```(1 44 25 24)(2 13 26 45)(3 46 27 14)(4 15 28 47)(5 48 29 16)(6 17 30 37)(7 38 31 18)(8 19 32 39)(9 40 33 20)(10 21 34 41)(11 42 35 22)(12 23 36 43)
(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 36 25 12)(2 11 26 35)(3 34 27 10)(4 9 28 33)(5 32 29 8)(6 7 30 31)(13 22 45 42)(14 41 46 21)(15 20 47 40)(16 39 48 19)(17 18 37 38)(23 24 43 44)```

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

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

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

Matrix representation of C4.D12 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 5 0 0 0 0 8
,
 6 10 0 0 3 3 0 0 0 0 0 1 0 0 1 0
,
 3 3 0 0 6 10 0 0 0 0 0 1 0 0 12 0
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[6,3,0,0,10,3,0,0,0,0,0,1,0,0,1,0],[3,6,0,0,3,10,0,0,0,0,0,12,0,0,1,0] >;`

C4.D12 in GAP, Magma, Sage, TeX

`C_4.D_{12}`
`% in TeX`

`G:=Group("C4.D12");`
`// GroupNames label`

`G:=SmallGroup(96,104);`
`// by ID`

`G=gap.SmallGroup(96,104);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,122,2309]);`
`// Polycyclic`

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

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