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## G = Q8.7D6order 96 = 25·3

### 2nd non-split extension by Q8 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8.7D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — D4⋊2S3 — Q8.7D6
 Lower central C3 — C6 — C12 — Q8.7D6
 Upper central C1 — C2 — C4 — SD16

Generators and relations for Q8.7D6
G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 154 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, Q8.7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S3×D4, Q8.7D6

Character table of Q8.7D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 12A 12B 24A 24B size 1 1 4 6 12 2 2 3 3 4 12 2 8 2 2 6 6 4 8 4 4 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 0 0 -1 2 0 0 2 0 -1 1 -2 -2 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 0 -2 0 2 -2 2 2 0 0 2 0 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 -1 2 0 0 -2 0 -1 -1 -2 -2 0 0 -1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 0 0 -1 2 0 0 2 0 -1 -1 2 2 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 0 0 -1 2 0 0 -2 0 -1 1 2 2 0 0 -1 1 -1 -1 orthogonal lifted from D6 ρ14 2 2 0 2 0 2 -2 -2 -2 0 0 2 0 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 √-2 -√-2 √2 -√2 0 0 √-2 -√-2 complex lifted from C4○D8 ρ16 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 √-2 -√-2 -√2 √2 0 0 √-2 -√-2 complex lifted from C4○D8 ρ17 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 -√-2 √-2 -√2 √2 0 0 -√-2 √-2 complex lifted from C4○D8 ρ18 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 -√-2 √-2 √2 -√2 0 0 -√-2 √-2 complex lifted from C4○D8 ρ19 4 4 0 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 0 -2 0 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 0 -√-2 √-2 complex faithful, Schur index 2 ρ21 4 -4 0 0 0 -2 0 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 0 √-2 -√-2 complex faithful, Schur index 2

Smallest permutation representation of Q8.7D6
On 48 points
Generators in S48
```(1 23 37 33)(2 34 38 24)(3 19 39 35)(4 36 40 20)(5 21 41 31)(6 32 42 22)(7 43 27 13)(8 14 28 44)(9 45 29 15)(10 16 30 46)(11 47 25 17)(12 18 26 48)
(1 43 37 13)(2 28 38 8)(3 45 39 15)(4 30 40 10)(5 47 41 17)(6 26 42 12)(7 33 27 23)(9 35 29 19)(11 31 25 21)(14 24 44 34)(16 20 46 36)(18 22 48 32)
(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 18 37 48)(2 47 38 17)(3 16 39 46)(4 45 40 15)(5 14 41 44)(6 43 42 13)(7 22 27 32)(8 31 28 21)(9 20 29 36)(10 35 30 19)(11 24 25 34)(12 33 26 23)```

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

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

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

Matrix representation of Q8.7D6 in GL4(𝔽73) generated by

 0 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 27 0 0 27 0 0 0 0 0 72 0 0 0 0 72
,
 16 16 0 0 16 57 0 0 0 0 0 1 0 0 72 1
,
 6 67 0 0 67 67 0 0 0 0 1 72 0 0 0 72
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,27,0,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,16,57,0,0,0,0,0,72,0,0,1,1],[6,67,0,0,67,67,0,0,0,0,1,0,0,0,72,72] >;`

Q8.7D6 in GAP, Magma, Sage, TeX

`Q_8._7D_6`
`% in TeX`

`G:=Group("Q8.7D6");`
`// GroupNames label`

`G:=SmallGroup(96,123);`
`// by ID`

`G=gap.SmallGroup(96,123);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,362,116,86,297,159,69,2309]);`
`// Polycyclic`

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

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