Copied to
clipboard

## G = D12⋊6C22order 96 = 25·3

### 4th semidirect product of D12 and C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12⋊6C22
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — D12⋊6C22
 Lower central C3 — C6 — C12 — D12⋊6C22
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D126C22
G = < a,b,c,d | a12=b2=c2=d2=1, bab=a-1, ac=ca, dad=a7, cbc=a6b, dbd=a3b, cd=dc >

Subgroups: 162 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C22, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, D126C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, C2×C3⋊D4, D126C22

Character table of D126C22

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B size 1 1 2 4 4 12 2 2 2 12 2 2 2 4 4 4 4 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 -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 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 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 linear of order 2 ρ9 2 2 2 2 2 0 -1 2 2 0 -1 -1 -1 -1 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 -2 -2 0 -1 2 2 0 -1 -1 -1 1 1 1 1 0 0 -1 -1 orthogonal lifted from D6 ρ11 2 2 -2 2 -2 0 -1 -2 2 0 1 1 -1 -1 -1 1 1 0 0 -1 1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 0 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 2 0 0 0 2 -2 -2 0 2 2 2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ14 2 2 -2 -2 2 0 -1 -2 2 0 1 1 -1 1 1 -1 -1 0 0 -1 1 orthogonal lifted from D6 ρ15 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 -1 -√-3 √-3 -√-3 √-3 0 0 1 1 complex lifted from C3⋊D4 ρ16 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 -1 √-3 -√-3 √-3 -√-3 0 0 1 1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 √-3 -√-3 -√-3 √-3 0 0 1 -1 complex lifted from C3⋊D4 ρ18 2 2 -2 0 0 0 -1 2 -2 0 1 1 -1 -√-3 √-3 √-3 -√-3 0 0 1 -1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of D126C22
On 24 points - transitive group 24T118
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 8)(4 10)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(4,10)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)>;`

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

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

`G:=TransitiveGroup(24,118);`

Matrix representation of D126C22 in GL4(𝔽7) generated by

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

D126C22 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_6C_2^2`
`% in TeX`

`G:=Group("D12:6C2^2");`
`// GroupNames label`

`G:=SmallGroup(96,139);`
`// by ID`

`G=gap.SmallGroup(96,139);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,579,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c=a^6*b,d*b*d=a^3*b,c*d=d*c>;`
`// generators/relations`

Export

׿
×
𝔽