Copied to
clipboard

## G = Q8.14D12order 192 = 26·3

### 4th non-split extension by Q8 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8.14D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8○D12 — Q8.14D12
 Lower central C3 — C6 — C2×C12 — Q8.14D12
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for Q8.14D12
G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=c11 >

Subgroups: 416 in 142 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, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C24⋊C2, Dic12, C4.Dic3, C4⋊Dic3, D4.S3, C3⋊Q16, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, S3×Q8, C3×C4○D4, D4.10D4, C424S3, C12.47D4, C3×C4≀C2, C122Q8, C8.D6, Q8.14D6, Q8○D12, Q8.14D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.10D4, D6⋊D4, Q8.14D12

Character table of Q8.14D12

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B size 1 1 2 4 12 2 2 2 4 4 4 12 12 12 24 2 4 8 8 24 2 2 4 4 4 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 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 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 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 -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 1 -1 -1 linear of order 2 ρ9 2 2 2 -2 0 -1 2 2 -2 2 2 0 0 0 0 -1 -1 1 -2 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 0 2 2 -2 2 0 0 0 0 -2 0 0 2 -2 0 0 0 2 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 2 -2 -2 0 0 0 -2 0 2 0 2 2 0 0 0 -2 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 0 -1 2 2 -2 -2 -2 0 0 0 0 -1 -1 1 2 0 -1 -1 1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ13 2 2 2 2 0 -1 2 2 2 2 2 0 0 0 0 -1 -1 -1 2 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 0 0 2 -2 -2 0 0 0 2 0 -2 0 2 2 0 0 0 -2 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 2 2 -2 2 0 0 0 0 0 0 2 -2 -2 0 0 -2 -2 0 2 0 0 0 2 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 0 -1 2 2 2 -2 -2 0 0 0 0 -1 -1 -1 -2 0 -1 -1 1 -1 1 1 1 -1 1 1 orthogonal lifted from D6 ρ17 2 2 -2 2 0 2 2 -2 -2 0 0 0 0 0 0 2 -2 2 0 0 -2 -2 0 2 0 0 0 -2 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 -2 2 -2 2 0 0 0 0 2 0 0 2 -2 0 0 0 2 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 0 -1 2 -2 -2 0 0 0 0 0 0 -1 1 -1 0 0 1 1 -√3 -1 √3 -√3 √3 1 -√3 √3 orthogonal lifted from D12 ρ20 2 2 -2 -2 0 -1 2 -2 2 0 0 0 0 0 0 -1 1 1 0 0 1 1 √3 -1 -√3 √3 -√3 -1 -√3 √3 orthogonal lifted from D12 ρ21 2 2 -2 2 0 -1 2 -2 -2 0 0 0 0 0 0 -1 1 -1 0 0 1 1 √3 -1 -√3 √3 -√3 1 √3 -√3 orthogonal lifted from D12 ρ22 2 2 -2 -2 0 -1 2 -2 2 0 0 0 0 0 0 -1 1 1 0 0 1 1 -√3 -1 √3 -√3 √3 -1 √3 -√3 orthogonal lifted from D12 ρ23 4 4 4 0 0 -2 -4 -4 0 0 0 0 0 0 0 -2 -2 0 0 0 2 2 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 -2 -4 4 0 0 0 0 0 0 0 -2 2 0 0 0 -2 -2 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 4 0 0 0 2 -2 0 0 0 0 -4 0 0 0 0 0 0 -2 0 2 2 -2 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ26 4 -4 0 0 0 4 0 0 0 -2 2 0 0 0 0 -4 0 0 0 0 0 0 2 0 -2 -2 2 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ27 4 -4 0 0 0 -2 0 0 0 -2 2 0 0 0 0 2 0 0 0 0 -2√3 2√3 -1+√3 0 1+√3 1-√3 -1-√3 0 0 0 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 0 -2 0 0 0 2 -2 0 0 0 0 2 0 0 0 0 -2√3 2√3 1-√3 0 -1-√3 -1+√3 1+√3 0 0 0 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 0 -2 0 0 0 2 -2 0 0 0 0 2 0 0 0 0 2√3 -2√3 1+√3 0 -1+√3 -1-√3 1-√3 0 0 0 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 0 -2 0 0 0 -2 2 0 0 0 0 2 0 0 0 0 2√3 -2√3 -1-√3 0 1-√3 1+√3 -1+√3 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8.14D12
On 48 points
Generators in S48
```(1 7 13 19)(2 20 14 8)(3 9 15 21)(4 22 16 10)(5 11 17 23)(6 24 18 12)(25 43 37 31)(26 32 38 44)(27 45 39 33)(28 34 40 46)(29 47 41 35)(30 36 42 48)
(1 4 13 16)(2 11 14 23)(3 6 15 18)(5 8 17 20)(7 10 19 22)(9 12 21 24)(25 34 37 46)(26 29 38 41)(27 36 39 48)(28 31 40 43)(30 33 42 45)(32 35 44 47)
(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 44 13 32)(2 31 14 43)(3 42 15 30)(4 29 16 41)(5 40 17 28)(6 27 18 39)(7 38 19 26)(8 25 20 37)(9 36 21 48)(10 47 22 35)(11 34 23 46)(12 45 24 33)```

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

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

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

Matrix representation of Q8.14D12 in GL4(𝔽73) generated by

 66 59 0 0 14 7 0 0 66 59 7 14 14 7 59 66
,
 1 0 71 0 0 1 0 71 1 0 72 0 0 1 0 72
,
 72 72 2 2 1 0 71 0 69 3 1 1 70 66 72 0
,
 8 34 0 0 26 65 0 0 68 7 18 20 12 5 2 55
`G:=sub<GL(4,GF(73))| [66,14,66,14,59,7,59,7,0,0,7,59,0,0,14,66],[1,0,1,0,0,1,0,1,71,0,72,0,0,71,0,72],[72,1,69,70,72,0,3,66,2,71,1,72,2,0,1,0],[8,26,68,12,34,65,7,5,0,0,18,2,0,0,20,55] >;`

Q8.14D12 in GAP, Magma, Sage, TeX

`Q_8._{14}D_{12}`
`% in TeX`

`G:=Group("Q8.14D12");`
`// GroupNames label`

`G:=SmallGroup(192,385);`
`// by ID`

`G=gap.SmallGroup(192,385);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,136,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=c^11>;`
`// generators/relations`

Export

׿
×
𝔽