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G = Q8⋊5D12order 192 = 26·3

3rd semidirect product of Q8 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8⋊5D12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — D4○D12 — Q8⋊5D12
 Lower central C3 — C6 — C2×C12 — Q8⋊5D12
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for Q85D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 672 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], Dic3, C12 [×2], C12 [×3], D6 [×10], C2×C6, C2×C6, C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4 [×8], C4○D4, C4○D4 [×3], C3⋊C8, C24, Dic6, C4×S3 [×3], D12, D12 [×10], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3 [×2], C22×S3 [×3], C4.D4, C4≀C2, C4≀C2, C41D4, C8⋊C22 [×2], 2+ 1+4, C24⋊C2, D24, C4.Dic3, D4⋊S3, Q82S3, C4×C12, C3×M4(2), C2×D12 [×2], C2×D12 [×3], C4○D12, C4○D12, S3×D4 [×3], Q83S3, C3×C4○D4, D44D4, C424S3, C12.46D4, C3×C4≀C2, C4⋊D12, C8⋊D6, D4⋊D6, D4○D12, Q85D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D44D4, D6⋊D4, Q85D12

Character table of Q85D12

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B size 1 1 2 4 12 12 12 24 2 2 2 4 4 4 12 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 0 0 0 -1 2 2 -2 -2 2 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 0 2 -2 0 2 -2 -2 0 0 0 0 2 2 0 0 0 -2 -2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 0 2 2 -2 0 0 2 0 2 -2 -2 0 0 -2 -2 0 0 0 0 2 2 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 2 0 0 0 2 -2 2 0 0 0 -2 2 -2 0 0 0 2 2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 0 0 0 0 2 2 -2 0 0 -2 0 2 -2 2 0 0 -2 -2 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 -2 0 0 0 2 -2 2 0 0 0 2 2 -2 0 0 0 2 2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 -2 0 0 0 0 -1 2 2 2 2 -2 0 -1 -1 1 -2 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ16 2 2 2 0 0 -2 2 0 2 -2 -2 0 0 0 0 2 2 0 0 0 -2 -2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 -2 0 0 0 0 -1 2 2 -2 -2 -2 0 -1 -1 1 2 0 -1 -1 1 1 1 1 -1 1 -1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 0 0 0 0 -1 2 2 2 2 2 0 -1 -1 -1 2 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 -2 2 0 0 0 0 -1 2 -2 0 0 -2 0 -1 1 -1 0 0 1 1 √3 √3 -√3 -√3 -1 1 √3 -√3 orthogonal lifted from D12 ρ20 2 2 -2 -2 0 0 0 0 -1 2 -2 0 0 2 0 -1 1 1 0 0 1 1 √3 √3 -√3 -√3 -1 -1 -√3 √3 orthogonal lifted from D12 ρ21 2 2 -2 2 0 0 0 0 -1 2 -2 0 0 -2 0 -1 1 -1 0 0 1 1 -√3 -√3 √3 √3 -1 1 -√3 √3 orthogonal lifted from D12 ρ22 2 2 -2 -2 0 0 0 0 -1 2 -2 0 0 2 0 -1 1 1 0 0 1 1 -√3 -√3 √3 √3 -1 -1 √3 -√3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 4 0 0 -2 2 0 0 -4 0 0 0 0 0 0 2 -2 -2 2 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 -2 -2 0 0 0 0 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 2 2 0 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 4 0 0 2 -2 0 0 -4 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ27 4 -4 0 0 0 0 0 0 -2 0 0 2 -2 0 0 2 0 0 0 0 2√3 -2√3 1+√3 -1-√3 -1+√3 1-√3 0 0 0 0 orthogonal faithful ρ28 4 -4 0 0 0 0 0 0 -2 0 0 -2 2 0 0 2 0 0 0 0 -2√3 2√3 -1+√3 1-√3 1+√3 -1-√3 0 0 0 0 orthogonal faithful ρ29 4 -4 0 0 0 0 0 0 -2 0 0 -2 2 0 0 2 0 0 0 0 2√3 -2√3 -1-√3 1+√3 1-√3 -1+√3 0 0 0 0 orthogonal faithful ρ30 4 -4 0 0 0 0 0 0 -2 0 0 2 -2 0 0 2 0 0 0 0 -2√3 2√3 1-√3 -1+√3 -1-√3 1+√3 0 0 0 0 orthogonal faithful

Permutation representations of Q85D12
On 24 points - transitive group 24T365
Generators in S24
```(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 19 16)(14 23 20 17)(15 24 21 18)
(1 22 4 16)(2 20 5 14)(3 18 6 24)(7 19 10 13)(8 17 11 23)(9 15 12 21)
(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 6)(2 5)(3 4)(7 9)(10 12)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)```

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

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

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

`G:=TransitiveGroup(24,365);`

Matrix representation of Q85D12 in GL4(𝔽73) generated by

 7 14 0 0 59 66 0 0 0 0 66 59 0 0 14 7
,
 0 0 1 0 0 0 0 1 72 0 0 0 0 72 0 0
,
 72 72 0 0 1 0 0 0 0 0 7 66 0 0 7 14
,
 72 72 0 0 0 1 0 0 0 0 7 66 0 0 59 66
`G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[72,1,0,0,72,0,0,0,0,0,7,7,0,0,66,14],[72,0,0,0,72,1,0,0,0,0,7,59,0,0,66,66] >;`

Q85D12 in GAP, Magma, Sage, TeX

`Q_8\rtimes_5D_{12}`
`% in TeX`

`G:=Group("Q8:5D12");`
`// GroupNames label`

`G:=SmallGroup(192,381);`
`// by ID`

`G=gap.SmallGroup(192,381);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,851,102,6278]);`
`// Polycyclic`

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

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