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## G = M4(2).Q8order 128 = 27

### 1st non-split extension by M4(2) of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — M4(2).Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — M4(2)⋊C4 — M4(2).Q8
 Lower central C1 — C2 — C22×C4 — M4(2).Q8
 Upper central C1 — C22 — C22×C4 — M4(2).Q8
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).Q8

Generators and relations for M4(2).Q8
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6bc2, bab=a5, cac-1=ab, dad-1=a3, cbc-1=dbd-1=a4b, dcd-1=a2bc3 >

Subgroups: 168 in 88 conjugacy classes, 42 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×4], C2×C4⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C4.C42, C22.C42 [×2], C42.6C22, C2×C2.D8, M4(2)⋊C4 [×2], M4(2).Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C23.81C23, D4.4D4, D4.5D4, M4(2).Q8

Character table of M4(2).Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 8 8 4 4 4 4 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 -2 -2 2 2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -2 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 2 -2 -2 2 2 -2 0 -2i 0 0 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 -2 2 -2 2 -2 2 0 0 -2i 0 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 -2 2 -2 2 0 0 2i 0 0 -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 2 -2 -2 2 2 -2 0 2i 0 0 -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of M4(2).Q8
On 64 points
Generators in S64
```(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)
(1 28 57 52 5 32 61 56)(2 25 62 53 6 29 58 49)(3 26 59 50 7 30 63 54)(4 31 64 51 8 27 60 55)(9 19 33 45 13 23 37 41)(10 24 38 46 14 20 34 42)(11 17 35 43 15 21 39 47)(12 22 40 44 16 18 36 48)
(1 10 63 40)(2 13 64 35)(3 16 57 38)(4 11 58 33)(5 14 59 36)(6 9 60 39)(7 12 61 34)(8 15 62 37)(17 27 45 53)(18 30 46 56)(19 25 47 51)(20 28 48 54)(21 31 41 49)(22 26 42 52)(23 29 43 55)(24 32 44 50)```

`G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64), (1,28,57,52,5,32,61,56)(2,25,62,53,6,29,58,49)(3,26,59,50,7,30,63,54)(4,31,64,51,8,27,60,55)(9,19,33,45,13,23,37,41)(10,24,38,46,14,20,34,42)(11,17,35,43,15,21,39,47)(12,22,40,44,16,18,36,48), (1,10,63,40)(2,13,64,35)(3,16,57,38)(4,11,58,33)(5,14,59,36)(6,9,60,39)(7,12,61,34)(8,15,62,37)(17,27,45,53)(18,30,46,56)(19,25,47,51)(20,28,48,54)(21,31,41,49)(22,26,42,52)(23,29,43,55)(24,32,44,50)>;`

`G:=Group( (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64), (1,28,57,52,5,32,61,56)(2,25,62,53,6,29,58,49)(3,26,59,50,7,30,63,54)(4,31,64,51,8,27,60,55)(9,19,33,45,13,23,37,41)(10,24,38,46,14,20,34,42)(11,17,35,43,15,21,39,47)(12,22,40,44,16,18,36,48), (1,10,63,40)(2,13,64,35)(3,16,57,38)(4,11,58,33)(5,14,59,36)(6,9,60,39)(7,12,61,34)(8,15,62,37)(17,27,45,53)(18,30,46,56)(19,25,47,51)(20,28,48,54)(21,31,41,49)(22,26,42,52)(23,29,43,55)(24,32,44,50) );`

`G=PermutationGroup([(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64)], [(1,28,57,52,5,32,61,56),(2,25,62,53,6,29,58,49),(3,26,59,50,7,30,63,54),(4,31,64,51,8,27,60,55),(9,19,33,45,13,23,37,41),(10,24,38,46,14,20,34,42),(11,17,35,43,15,21,39,47),(12,22,40,44,16,18,36,48)], [(1,10,63,40),(2,13,64,35),(3,16,57,38),(4,11,58,33),(5,14,59,36),(6,9,60,39),(7,12,61,34),(8,15,62,37),(17,27,45,53),(18,30,46,56),(19,25,47,51),(20,28,48,54),(21,31,41,49),(22,26,42,52),(23,29,43,55),(24,32,44,50)])`

Matrix representation of M4(2).Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 16 0 0 0 0 0 0 3 11 2 3 0 0 10 4 3 15 0 0 15 5 1 10 0 0 7 11 9 9
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 16 0 0 0 3 4 0 16
,
 0 5 0 0 0 0 7 0 0 0 0 0 0 0 5 5 6 6 0 0 11 5 11 6 0 0 4 4 3 3 0 0 11 2 14 4
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 3 4 0 15 0 0 16 0 2 0 0 0 10 11 0 16 0 0 2 6 4 14

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,3,10,15,7,0,0,11,4,5,11,0,0,2,3,1,9,0,0,3,15,10,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,3,0,0,0,1,0,4,0,0,0,0,16,0,0,0,0,0,0,16],[0,7,0,0,0,0,5,0,0,0,0,0,0,0,5,11,4,11,0,0,5,5,4,2,0,0,6,11,3,14,0,0,6,6,3,4],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,10,2,0,0,4,0,11,6,0,0,0,2,0,4,0,0,15,0,16,14] >;`

M4(2).Q8 in GAP, Magma, Sage, TeX

`M_4(2).Q_8`
`% in TeX`

`G:=Group("M4(2).Q8");`
`// GroupNames label`

`G:=SmallGroup(128,821);`
`// by ID`

`G=gap.SmallGroup(128,821);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,718,172,2028,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b*c^2,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*b*c^3>;`
`// generators/relations`

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