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G = M4(2).2Q8order 128 = 27

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

Generators and relations for M4(2).2Q8
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2bc2, bab=a5, cac-1=a5b, dad-1=a-1, cbc-1=dbd-1=a4b, dcd-1=a6bc3 >

Subgroups: 168 in 88 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2 [×2], 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 [×4], C2.D8 [×2], C2×C4⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C4.C42, C22.C42 [×2], C42.6C22, C2×C4.Q8, M4(2)⋊C4 [×2], M4(2).2Q8
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, D4.3D4 [×2], M4(2).2Q8

Character table of M4(2).2Q8

 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 -2√-2 2√-2 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 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 complex lifted from D4.3D4 ρ25 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 complex lifted from D4.3D4 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 complex lifted from D4.3D4

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

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

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

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

 0 15 0 0 0 0 9 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0
,
 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 0 0 16 0 0 0 0 0 0 16
,
 0 8 0 0 0 0 2 0 0 0 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 5 12 0 0 0 0 5 5 0 0
,
 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,5,5,0,0,0,0,12,5,0,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0] >;`

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

`M_4(2)._2Q_8`
`% in TeX`

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

`G:=SmallGroup(128,822);`
`// by ID`

`G=gap.SmallGroup(128,822);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,718,172,2028,1027,124]);`
`// Polycyclic`

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

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