Copied to
clipboard

## G = D12⋊18D4order 192 = 26·3

### 6th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊18D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — D4○D12 — D12⋊18D4
 Lower central C3 — C6 — C2×C12 — D12⋊18D4
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D1218D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a10b, dcd=c-1 >

Subgroups: 624 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C4.D4, C4≀C2, C41D4, C8⋊C22, C8⋊C22, 2+ 1+4, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C3×SD16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, Q83S3, C2×C3⋊D4, C6×D4, C3×C4○D4, D44D4, C12.46D4, D12⋊C4, Q83Dic3, D126C22, C123D4, C3×C8⋊C22, D4○D12, D1218D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D44D4, C232D6, D1218D4

Character table of D1218D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 24A 24B size 1 1 2 4 8 12 12 12 2 2 2 4 12 12 12 2 4 8 8 8 8 24 4 4 8 8 8 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 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 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 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 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 -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 -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 -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 -1 1 1 -1 1 1 linear of order 2 ρ9 2 2 2 2 2 0 0 0 -1 2 2 2 0 0 0 -1 -1 -1 -1 -1 2 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 0 0 0 0 2 -2 2 2 0 0 0 2 -2 -2 0 0 0 0 2 -2 2 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 -2 0 0 2 2 -2 0 2 0 0 2 -2 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 -2 0 0 0 -1 2 2 -2 0 0 0 -1 -1 1 1 1 2 0 -1 -1 1 -1 -1 orthogonal lifted from D6 ρ13 2 2 2 0 0 0 2 -2 2 -2 -2 0 0 0 0 2 2 0 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 0 0 0 -1 2 2 2 0 0 0 -1 -1 -1 1 1 -2 0 -1 -1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 2 -2 2 0 0 0 0 2 -2 -2 0 0 orthogonal lifted from D4 ρ16 2 2 2 -2 2 0 0 0 -1 2 2 -2 0 0 0 -1 -1 1 -1 -1 -2 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ17 2 2 2 0 0 0 -2 2 2 -2 -2 0 0 0 0 2 2 0 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 0 2 0 0 2 2 -2 0 -2 0 0 2 -2 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 0 0 0 -1 -2 2 2 0 0 0 -1 1 1 -√-3 √-3 0 0 -1 1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 0 0 0 -1 -2 2 -2 0 0 0 -1 1 -1 -√-3 √-3 0 0 -1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 0 0 -1 -2 2 2 0 0 0 -1 1 1 √-3 -√-3 0 0 -1 1 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 -2 2 0 0 0 0 -1 -2 2 -2 0 0 0 -1 1 -1 √-3 -√-3 0 0 -1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 -4 0 0 0 0 0 0 4 0 0 0 0 -2 2 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ24 4 4 4 0 0 0 0 0 -2 -4 -4 0 0 0 0 -2 -2 0 0 0 0 0 2 2 0 0 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 0 0 0 0 0 -2 4 -4 0 0 0 0 -2 2 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 4 0 0 0 0 2 -2 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ27 8 -8 0 0 0 0 0 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D1218D4
On 24 points - transitive group 24T366
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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)
(2 6)(3 11)(5 9)(8 12)(13 22 19 16)(14 15 20 21)(17 18 23 24)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 19)(14 18)(15 17)(20 24)(21 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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (2,6)(3,11)(5,9)(8,12)(13,22,19,16)(14,15,20,21)(17,18,23,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,19)(14,18)(15,17)(20,24)(21,23)>;`

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

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

`G:=TransitiveGroup(24,366);`

Matrix representation of D1218D4 in GL8(ℤ)

 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0
,
 0 0 0 0 -1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 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 0 0 0

`G:=sub<GL(8,Integers())| [0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0] >;`

D1218D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{18}D_4`
`% in TeX`

`G:=Group("D12:18D4");`
`// GroupNames label`

`G:=SmallGroup(192,757);`
`// by ID`

`G=gap.SmallGroup(192,757);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,570,1684,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽