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## G = C12.D4order 96 = 25·3

### 8th non-split extension by C12 of 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 — C12 — C2×C12 — C4.Dic3 — C12.D4
 Lower central C3 — C6 — C2×C6 — C12.D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C12.D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b3 >

Character table of C12.D4

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B size 1 1 2 4 4 2 2 2 2 2 2 4 4 4 4 12 12 12 12 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 i -i -i i -1 -1 linear of order 4 ρ6 1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -i i i -i -1 -1 linear of order 4 ρ7 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -i -i i i -1 -1 linear of order 4 ρ8 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 i i -i -i -1 -1 linear of order 4 ρ9 2 2 2 -2 -2 -1 2 2 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 2 -1 2 2 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 0 0 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ12 2 2 -2 0 0 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 2 -2 2 -1 -2 -2 -1 -1 -1 1 -1 1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 2 -2 -1 -2 -2 -1 -1 -1 -1 1 -1 1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 -2 0 0 -1 2 -2 1 1 -1 -√-3 √-3 √-3 -√-3 0 0 0 0 -1 1 complex lifted from C3⋊D4 ρ16 2 2 -2 0 0 -1 2 -2 1 1 -1 √-3 -√-3 -√-3 √-3 0 0 0 0 -1 1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 0 -1 -2 2 1 1 -1 √-3 √-3 -√-3 -√-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ18 2 2 -2 0 0 -1 -2 2 1 1 -1 -√-3 -√-3 √-3 √-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ20 4 -4 0 0 0 -2 0 0 -2√-3 2√-3 2 0 0 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 -2 0 0 2√-3 -2√-3 2 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C12.D4
On 24 points - transitive group 24T99
Generators in S24
```(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 13 10 16 7 19 4 22)(2 24 11 15 8 18 5 21)(3 23 12 14 9 17 6 20)
(1 16 10 13 7 22 4 19)(2 21 11 18 8 15 5 24)(3 14 12 23 9 20 6 17)```

`G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,10,16,7,19,4,22)(2,24,11,15,8,18,5,21)(3,23,12,14,9,17,6,20), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17)>;`

`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), (1,13,10,16,7,19,4,22)(2,24,11,15,8,18,5,21)(3,23,12,14,9,17,6,20), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17) );`

`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)], [(1,13,10,16,7,19,4,22),(2,24,11,15,8,18,5,21),(3,23,12,14,9,17,6,20)], [(1,16,10,13,7,22,4,19),(2,21,11,18,8,15,5,24),(3,14,12,23,9,20,6,17)]])`

`G:=TransitiveGroup(24,99);`

Matrix representation of C12.D4 in GL4(𝔽7) generated by

 1 0 6 2 2 1 0 1 5 5 5 4 4 3 5 0
,
 6 0 3 2 0 5 1 2 4 3 5 2 1 1 3 5
,
 2 3 0 1 4 2 4 1 4 3 5 5 1 1 3 5
`G:=sub<GL(4,GF(7))| [1,2,5,4,0,1,5,3,6,0,5,5,2,1,4,0],[6,0,4,1,0,5,3,1,3,1,5,3,2,2,2,5],[2,4,4,1,3,2,3,1,0,4,5,3,1,1,5,5] >;`

C12.D4 in GAP, Magma, Sage, TeX

`C_{12}.D_4`
`% in TeX`

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

`G:=SmallGroup(96,40);`
`// by ID`

`G=gap.SmallGroup(96,40);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,86,579,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^3>;`
`// generators/relations`

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