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## G = C6.D4order 48 = 24·3

### 7th non-split extension by C6 of D4 acting via D4/C22=C2

Aliases: C6.11D4, C23.2S3, C22.7D6, C222Dic3, (C2×C6)⋊2C4, C6.9(C2×C4), C32(C22⋊C4), (C2×Dic3)⋊2C2, C2.3(C3⋊D4), (C22×C6).2C2, (C2×C6).7C22, C2.5(C2×Dic3), SmallGroup(48,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C6.D4
 Lower central C3 — C6 — C6.D4
 Upper central C1 — C22 — C23

Generators and relations for C6.D4
G = < a,b,c | a6=b4=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >

Character table of C6.D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G size 1 1 1 1 2 2 2 6 6 6 6 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 -i -i i i 1 -1 -1 -1 -1 1 1 linear of order 4 ρ6 1 -1 1 -1 -1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 -1 1 -1 -1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 1 -1 1 i i -i -i 1 -1 -1 -1 -1 1 1 linear of order 4 ρ9 2 2 2 2 2 2 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -2 -2 -1 0 0 0 0 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 -2 -2 2 0 0 2 0 0 0 0 -2 0 0 2 -2 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 0 2 0 0 0 0 -2 0 0 -2 2 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 -2 2 -1 0 0 0 0 -1 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 -2 2 -2 2 -2 -1 0 0 0 0 -1 1 1 1 1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 0 0 -1 0 0 0 0 1 -√-3 √-3 -1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ16 2 2 -2 -2 0 0 -1 0 0 0 0 1 √-3 -√-3 1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ17 2 -2 -2 2 0 0 -1 0 0 0 0 1 √-3 -√-3 -1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 0 0 0 0 1 -√-3 √-3 1 -1 -√-3 √-3 complex lifted from C3⋊D4

Permutation representations of C6.D4
On 24 points - transitive group 24T44
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 23 14 10)(2 22 15 9)(3 21 16 8)(4 20 17 7)(5 19 18 12)(6 24 13 11)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 17 10 14)(8 16 11 13)(9 15 12 18)```

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

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

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

`G:=TransitiveGroup(24,44);`

Matrix representation of C6.D4 in GL3(𝔽13) generated by

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

C6.D4 in GAP, Magma, Sage, TeX

`C_6.D_4`
`% in TeX`

`G:=Group("C6.D4");`
`// GroupNames label`

`G:=SmallGroup(48,19);`
`// by ID`

`G=gap.SmallGroup(48,19);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,804]);`
`// Polycyclic`

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

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