Copied to
clipboard

## G = D12.5D4order 192 = 26·3

### 5th non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.5D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.15D6 — D12.5D4
 Lower central C3 — C6 — C2×C12 — D12.5D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D12.5D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a9b, dcd-1=a9c3 >

Subgroups: 496 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C24⋊C2, D24, C4×Dic3, D6⋊C4, C3×M4(2), C2×D12, C4○D12, C4○D12, S3×Q8, Q83S3, C6×Q8, D4.8D4, D12⋊C4, C3×C4.10D4, C8⋊D6, C12.23D4, Q8.15D6, D12.5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.8D4, D6⋊D4, D12.5D4

Character table of D12.5D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 2 12 12 24 2 2 2 4 4 12 12 12 12 2 4 8 8 4 4 8 8 8 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 0 0 0 -1 2 2 -2 -2 0 0 0 0 -1 -1 2 -2 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 0 0 0 2 -2 -2 2 -2 0 0 0 0 2 2 0 0 -2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -2 -2 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 0 0 0 -1 2 2 -2 -2 0 0 0 0 -1 -1 -2 2 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 2 0 2 -2 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 0 0 2 -2 -2 -2 2 0 0 0 0 2 2 0 0 -2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 0 2 0 2 2 -2 0 0 0 -2 0 0 2 -2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 -2 2 0 0 2 -2 2 0 0 0 0 -2 0 2 -2 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 0 -2 0 2 2 -2 0 0 0 2 0 0 2 -2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 0 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 2 0 0 0 -1 -2 -2 2 -2 0 0 0 0 -1 -1 0 0 1 1 -1 1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ20 2 2 2 0 0 0 -1 -2 -2 2 -2 0 0 0 0 -1 -1 0 0 1 1 -1 1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ21 2 2 2 0 0 0 -1 -2 -2 -2 2 0 0 0 0 -1 -1 0 0 1 1 1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ22 2 2 2 0 0 0 -1 -2 -2 -2 2 0 0 0 0 -1 -1 0 0 1 1 1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ23 4 4 -4 0 0 0 -2 -4 4 0 0 0 0 0 0 -2 2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 -2 4 -4 0 0 0 0 0 0 -2 2 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 4 0 0 0 0 -2i 0 0 2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ26 4 -4 0 0 0 0 4 0 0 0 0 2i 0 0 -2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ27 8 -8 0 0 0 0 -4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of D12.5D4
On 48 points
Generators in S48
```(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 16 10 19 7 22 4 13)(2 23 11 14 8 17 5 20)(3 18 12 21 9 24 6 15)(25 43 34 46 31 37 28 40)(26 38 35 41 32 44 29 47)(27 45 36 48 33 39 30 42)
(1 46 10 43 7 40 4 37)(2 47 11 44 8 41 5 38)(3 48 12 45 9 42 6 39)(13 34 22 31 19 28 16 25)(14 35 23 32 20 29 17 26)(15 36 24 33 21 30 18 27)```

`G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,16,10,19,7,22,4,13)(2,23,11,14,8,17,5,20)(3,18,12,21,9,24,6,15)(25,43,34,46,31,37,28,40)(26,38,35,41,32,44,29,47)(27,45,36,48,33,39,30,42), (1,46,10,43,7,40,4,37)(2,47,11,44,8,41,5,38)(3,48,12,45,9,42,6,39)(13,34,22,31,19,28,16,25)(14,35,23,32,20,29,17,26)(15,36,24,33,21,30,18,27)>;`

`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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,16,10,19,7,22,4,13)(2,23,11,14,8,17,5,20)(3,18,12,21,9,24,6,15)(25,43,34,46,31,37,28,40)(26,38,35,41,32,44,29,47)(27,45,36,48,33,39,30,42), (1,46,10,43,7,40,4,37)(2,47,11,44,8,41,5,38)(3,48,12,45,9,42,6,39)(13,34,22,31,19,28,16,25)(14,35,23,32,20,29,17,26)(15,36,24,33,21,30,18,27) );`

`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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,16,10,19,7,22,4,13),(2,23,11,14,8,17,5,20),(3,18,12,21,9,24,6,15),(25,43,34,46,31,37,28,40),(26,38,35,41,32,44,29,47),(27,45,36,48,33,39,30,42)], [(1,46,10,43,7,40,4,37),(2,47,11,44,8,41,5,38),(3,48,12,45,9,42,6,39),(13,34,22,31,19,28,16,25),(14,35,23,32,20,29,17,26),(15,36,24,33,21,30,18,27)]])`

Matrix representation of D12.5D4 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 46 0 65 56 0 0 0 27 0 27 0 0 0 0 27 60 0 0 0 0 0 46
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 27 0 27 0 0 46 0 65 56 0 0 0 0 56 37 0 0 0 0 8 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 46 0 0 0 0 0 0 0 17 51 0 0 0 0 65 56
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 27 0 8 17 0 0 60 25 17 3 0 0 19 0 65 56

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,65,0,27,0,0,0,56,27,60,46],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,65,56,8,0,0,27,56,37,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0,0,0,0,0,0,0,17,65,0,0,1,0,51,56],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,60,19,0,0,72,0,25,0,0,0,0,8,17,65,0,0,0,17,3,56] >;`

D12.5D4 in GAP, Magma, Sage, TeX

`D_{12}._5D_4`
`% in TeX`

`G:=Group("D12.5D4");`
`// GroupNames label`

`G:=SmallGroup(192,312);`
`// by ID`

`G=gap.SmallGroup(192,312);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,570,136,1684,438,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^9*c^3>;`
`// generators/relations`

Export

׿
×
𝔽