Copied to
clipboard

## G = C6.D12order 144 = 24·32

### 6th non-split extension by C6 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6.D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C6.D12
 Lower central C32 — C3×C6 — C6.D12
 Upper central C1 — C22

Generators and relations for C6.D12
G = < a,b,c | a6=b12=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 336 in 84 conjugacy classes, 28 normal (8 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C3×Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, C6×Dic3, C22×C3⋊S3, C6.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, S32, D6⋊C4, C6.D6, C3⋊D12, C6.D12

Character table of C6.D12

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 18 18 2 2 4 6 6 6 6 2 2 2 2 2 2 4 4 4 6 6 6 6 6 6 6 6 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 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 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 -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 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 1 1 -i -i i i 1 -1 -1 -1 -1 1 1 -1 -1 i i i -i -i -i i -i linear of order 4 ρ6 1 1 -1 -1 -1 1 1 1 1 -i i i -i 1 -1 -1 -1 -1 1 1 -1 -1 -i i i i -i -i -i i linear of order 4 ρ7 1 1 -1 -1 1 -1 1 1 1 i i -i -i 1 -1 -1 -1 -1 1 1 -1 -1 -i -i -i i i i -i i linear of order 4 ρ8 1 1 -1 -1 -1 1 1 1 1 i -i -i i 1 -1 -1 -1 -1 1 1 -1 -1 i -i -i -i i i i -i linear of order 4 ρ9 2 2 2 2 0 0 -1 2 -1 0 -2 0 -2 2 -1 -1 2 2 -1 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ10 2 -2 2 -2 0 0 2 2 2 0 0 0 0 -2 2 -2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 2 -1 -1 -2 0 -2 0 -1 2 2 -1 -1 2 -1 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ12 2 2 2 2 0 0 2 -1 -1 2 0 2 0 -1 2 2 -1 -1 2 -1 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ13 2 -2 -2 2 0 0 2 2 2 0 0 0 0 -2 -2 2 2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 -1 2 -1 0 2 0 2 2 -1 -1 2 2 -1 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ15 2 -2 2 -2 0 0 -1 2 -1 0 0 0 0 -2 -1 1 -2 2 1 1 1 -1 √3 0 0 √3 0 0 -√3 -√3 orthogonal lifted from D12 ρ16 2 -2 -2 2 0 0 2 -1 -1 0 0 0 0 1 -2 2 -1 1 -2 1 -1 1 0 -√3 √3 0 √3 -√3 0 0 orthogonal lifted from D12 ρ17 2 -2 2 -2 0 0 -1 2 -1 0 0 0 0 -2 -1 1 -2 2 1 1 1 -1 -√3 0 0 -√3 0 0 √3 √3 orthogonal lifted from D12 ρ18 2 -2 -2 2 0 0 2 -1 -1 0 0 0 0 1 -2 2 -1 1 -2 1 -1 1 0 √3 -√3 0 -√3 √3 0 0 orthogonal lifted from D12 ρ19 2 -2 -2 2 0 0 -1 2 -1 0 0 0 0 -2 1 -1 2 -2 1 1 -1 1 -√-3 0 0 √-3 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ20 2 -2 2 -2 0 0 2 -1 -1 0 0 0 0 1 2 -2 1 -1 -2 1 1 -1 0 -√-3 √-3 0 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ21 2 -2 -2 2 0 0 -1 2 -1 0 0 0 0 -2 1 -1 2 -2 1 1 -1 1 √-3 0 0 -√-3 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 -2 2 -2 0 0 2 -1 -1 0 0 0 0 1 2 -2 1 -1 -2 1 1 -1 0 √-3 -√-3 0 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ23 2 2 -2 -2 0 0 2 -1 -1 -2i 0 2i 0 -1 -2 -2 1 1 2 -1 1 1 0 -i -i 0 i i 0 0 complex lifted from C4×S3 ρ24 2 2 -2 -2 0 0 2 -1 -1 2i 0 -2i 0 -1 -2 -2 1 1 2 -1 1 1 0 i i 0 -i -i 0 0 complex lifted from C4×S3 ρ25 2 2 -2 -2 0 0 -1 2 -1 0 -2i 0 2i 2 1 1 -2 -2 -1 -1 1 1 -i 0 0 i 0 0 -i i complex lifted from C4×S3 ρ26 2 2 -2 -2 0 0 -1 2 -1 0 2i 0 -2i 2 1 1 -2 -2 -1 -1 1 1 i 0 0 -i 0 0 i -i complex lifted from C4×S3 ρ27 4 4 -4 -4 0 0 -2 -2 1 0 0 0 0 -2 2 2 2 2 -2 1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C6.D6 ρ28 4 -4 4 -4 0 0 -2 -2 1 0 0 0 0 2 -2 2 2 -2 2 -1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ29 4 4 4 4 0 0 -2 -2 1 0 0 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ30 4 -4 -4 4 0 0 -2 -2 1 0 0 0 0 2 2 -2 -2 2 2 -1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12

Permutation representations of C6.D12
On 24 points - transitive group 24T235
Generators in S24
```(1 19 5 23 9 15)(2 16 10 24 6 20)(3 21 7 13 11 17)(4 18 12 14 8 22)
(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 9)(2 18)(3 7)(4 16)(6 14)(8 24)(10 22)(12 20)(13 17)(19 23)```

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

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

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

`G:=TransitiveGroup(24,235);`

Matrix representation of C6.D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 12 1

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

C6.D12 in GAP, Magma, Sage, TeX

`C_6.D_{12}`
`% in TeX`

`G:=Group("C6.D12");`
`// GroupNames label`

`G:=SmallGroup(144,65);`
`// by ID`

`G=gap.SmallGroup(144,65);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,79,490,3461]);`
`// Polycyclic`

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

Export

׿
×
𝔽