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G = Q8.13D6order 96 = 25·3

3rd non-split extension by Q8 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8.13D6
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — Q8.13D6
 Lower central C3 — C6 — C12 — Q8.13D6
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for Q8.13D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >

Subgroups: 138 in 62 conjugacy classes, 29 normal (all 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, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, Q8.13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, C2×C3⋊D4, Q8.13D6

Character table of Q8.13D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E size 1 1 2 4 12 2 1 1 2 4 12 2 4 4 4 6 6 6 6 2 2 4 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 0 0 2 -2 -2 -2 0 0 2 2 0 0 0 0 0 0 -2 -2 0 0 -2 orthogonal lifted from D4 ρ10 2 2 2 -2 0 -1 2 2 2 -2 0 -1 -1 1 1 0 0 0 0 -1 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 0 -1 2 2 2 2 0 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 0 2 2 2 -2 0 0 2 -2 0 0 0 0 0 0 2 2 0 0 -2 orthogonal lifted from D4 ρ13 2 2 -2 2 0 -1 -2 -2 2 -2 0 -1 1 -1 -1 0 0 0 0 1 1 1 1 -1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 -1 -2 -2 2 2 0 -1 1 1 1 0 0 0 0 1 1 -1 -1 -1 orthogonal lifted from D6 ρ15 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 √-3 -√-3 0 0 0 0 1 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ16 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 -√-3 √-3 0 0 0 0 -1 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 √-3 -√-3 0 0 0 0 -1 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ18 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 -√-3 √-3 0 0 0 0 1 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ19 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 -√-2 -√2 √-2 √2 2i -2i 0 0 0 complex lifted from C4○D8 ρ20 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 √-2 √2 -√-2 -√2 2i -2i 0 0 0 complex lifted from C4○D8 ρ21 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 -√-2 √2 √-2 -√2 -2i 2i 0 0 0 complex lifted from C4○D8 ρ22 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 √-2 -√2 -√-2 √2 -2i 2i 0 0 0 complex lifted from C4○D8 ρ23 4 -4 0 0 0 -2 -4i 4i 0 0 0 2 0 0 0 0 0 0 0 2i -2i 0 0 0 complex faithful ρ24 4 -4 0 0 0 -2 4i -4i 0 0 0 2 0 0 0 0 0 0 0 -2i 2i 0 0 0 complex faithful

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

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

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

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

Matrix representation of Q8.13D6 in GL4(𝔽5) generated by

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

Q8.13D6 in GAP, Magma, Sage, TeX

`Q_8._{13}D_6`
`% in TeX`

`G:=Group("Q8.13D6");`
`// GroupNames label`

`G:=SmallGroup(96,157);`
`// by ID`

`G=gap.SmallGroup(96,157);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,579,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=1,b^2=c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;`
`// generators/relations`

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