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## G = C12.46D4order 96 = 25·3

### 3rd non-split extension by C12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.46D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12.46D4
 Lower central C3 — C6 — C2×C6 — C12.46D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C12.46D4
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Character table of C12.46D4

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 8A 8B 8C 8D 12A 12B 12C 24A 24B 24C 24D size 1 1 2 12 12 2 2 2 2 4 4 4 12 12 2 2 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 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 -i i i -i -1 -1 -1 i i -i -i linear of order 4 ρ6 1 1 1 1 -1 1 -1 -1 1 1 -i i -i i -1 -1 -1 i i -i -i linear of order 4 ρ7 1 1 1 -1 1 1 -1 -1 1 1 i -i -i i -1 -1 -1 -i -i i i linear of order 4 ρ8 1 1 1 1 -1 1 -1 -1 1 1 i -i i -i -1 -1 -1 -i -i i i linear of order 4 ρ9 2 2 2 0 0 -1 2 2 -1 -1 -2 -2 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 0 0 2 2 -2 2 -2 0 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 -1 2 2 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 0 2 -2 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 -1 -2 2 -1 1 0 0 0 0 1 1 -1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ14 2 2 -2 0 0 -1 -2 2 -1 1 0 0 0 0 1 1 -1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ15 2 2 2 0 0 -1 -2 -2 -1 -1 -2i 2i 0 0 1 1 1 -i -i i i complex lifted from C4×S3 ρ16 2 2 2 0 0 -1 -2 -2 -1 -1 2i -2i 0 0 1 1 1 i i -i -i complex lifted from C4×S3 ρ17 2 2 -2 0 0 -1 2 -2 -1 1 0 0 0 0 -1 -1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 0 0 -1 2 -2 -1 1 0 0 0 0 -1 -1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 4 0 0 -4 0 0 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 0 0 0 0 0 -2√3 2√3 0 0 0 0 0 orthogonal faithful ρ21 4 -4 0 0 0 -2 0 0 2 0 0 0 0 0 2√3 -2√3 0 0 0 0 0 orthogonal faithful

Permutation representations of C12.46D4
On 24 points - transitive group 24T105
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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)
(1 11 22)(2 12 23)(3 13 24)(4 14 17)(5 15 18)(6 16 19)(7 9 20)(8 10 21)
(2 6)(3 7)(9 24)(10 21)(11 22)(12 19)(13 20)(14 17)(15 18)(16 23)```

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

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

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

`G:=TransitiveGroup(24,105);`

Matrix representation of C12.46D4 in GL6(𝔽73)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 11 68 21 3 0 0 0 72 0 0 0 0 43 23 7 5
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 17 52 0 72
,
 36 28 0 0 0 0 28 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 11 68 21 3 0 0 22 7 48 52

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,43,0,0,0,68,72,23,0,0,1,21,0,7,0,0,0,3,0,5],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,17,0,0,0,1,0,52,0,0,0,0,72,0,0,0,0,0,0,72],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,11,22,0,0,1,0,68,7,0,0,0,0,21,48,0,0,0,0,3,52] >;`

C12.46D4 in GAP, Magma, Sage, TeX

`C_{12}._{46}D_4`
`% in TeX`

`G:=Group("C12.46D4");`
`// GroupNames label`

`G:=SmallGroup(96,30);`
`// by ID`

`G=gap.SmallGroup(96,30);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,86,297,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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