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G = Q8⋊3D6order 96 = 25·3

2nd semidirect product of Q8 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8⋊3D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×D4 — Q8⋊3D6
 Lower central C3 — C6 — C12 — Q8⋊3D6
 Upper central C1 — C2 — C4 — SD16

Generators and relations for Q83D6
G = < a,b,c,d | a4=c6=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 202 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, Dic3, C12, C12, D6, D6 [×4], C2×C6, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, Q83D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C22×S3, C8⋊C22, S3×D4, Q83D6

Character table of Q83D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 8A 8B 12A 12B 24A 24B size 1 1 4 6 12 12 2 2 4 6 2 8 4 12 4 8 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 0 2 0 0 2 -2 0 -2 2 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 0 -1 2 -2 0 -1 1 2 0 -1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 0 0 0 -1 2 -2 0 -1 -1 -2 0 -1 1 1 1 orthogonal lifted from D6 ρ12 2 2 0 -2 0 0 2 -2 0 2 2 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 0 -1 2 2 0 -1 -1 2 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 0 0 0 -1 2 2 0 -1 1 -2 0 -1 -1 1 1 orthogonal lifted from D6 ρ15 4 4 0 0 0 0 -2 -4 0 0 -2 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ16 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ17 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 √6 -√6 orthogonal faithful ρ18 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 -√6 √6 orthogonal faithful

Permutation representations of Q83D6
On 24 points - transitive group 24T140
Generators in S24
```(1 10 4 7)(2 8 5 11)(3 12 6 9)(13 23 20 16)(14 17 21 24)(15 19 22 18)
(1 19 4 18)(2 23 5 16)(3 21 6 14)(7 22 10 15)(8 13 11 20)(9 24 12 17)
(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 3)(4 6)(7 12)(8 11)(9 10)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)```

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

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

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

`G:=TransitiveGroup(24,140);`

Matrix representation of Q83D6 in GL4(𝔽5) generated by

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

Q83D6 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3D_6`
`% in TeX`

`G:=Group("Q8:3D6");`
`// GroupNames label`

`G:=SmallGroup(96,121);`
`// by ID`

`G=gap.SmallGroup(96,121);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,86,297,159,69,2309]);`
`// Polycyclic`

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

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