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## G = C12⋊D4order 96 = 25·3

### 1st semidirect product of C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C12⋊D4
 Lower central C3 — C2×C6 — C12⋊D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C12⋊D4
G = < a,b,c | a12=b4=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >

Subgroups: 266 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C4⋊D4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C12⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, C12⋊D4

Character table of C12⋊D4

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

Smallest permutation representation of C12⋊D4
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 22 33)(2 42 23 28)(3 37 24 35)(4 44 13 30)(5 39 14 25)(6 46 15 32)(7 41 16 27)(8 48 17 34)(9 43 18 29)(10 38 19 36)(11 45 20 31)(12 40 21 26)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 24)(12 23)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 48)```

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

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

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

Matrix representation of C12⋊D4 in GL4(𝔽13) generated by

 1 12 0 0 1 0 0 0 0 0 1 11 0 0 1 12
,
 3 7 0 0 6 10 0 0 0 0 12 0 0 0 12 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 12
`G:=sub<GL(4,GF(13))| [1,1,0,0,12,0,0,0,0,0,1,1,0,0,11,12],[3,6,0,0,7,10,0,0,0,0,12,12,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,12] >;`

C12⋊D4 in GAP, Magma, Sage, TeX

`C_{12}\rtimes D_4`
`% in TeX`

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

`G:=SmallGroup(96,102);`
`// by ID`

`G=gap.SmallGroup(96,102);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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