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## G = Q8⋊5M4(2)  order 128 = 27

### 3rd semidirect product of Q8 and M4(2) acting via M4(2)/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8⋊5M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×Q8 — Q8⋊5M4(2)
 Lower central C1 — C2 — C2×C4 — Q8⋊5M4(2)
 Upper central C1 — C2×C4 — C2×C42 — Q8⋊5M4(2)
 Jennings C1 — C22 — C22 — C42 — Q8⋊5M4(2)

Generators and relations for Q85M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >

Subgroups: 212 in 126 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×16], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×5], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8, C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C2×M4(2), C22×Q8, Q8⋊C8 [×4], C4×M4(2), C42.6C4, C2×C4×Q8, Q85M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×M4(2) [×2], C8.C22 [×2], C24.4C4, C23.38D4, C2×C4≀C2, Q85M4(2)

Smallest permutation representation of Q85M4(2)
On 64 points
Generators in S64
```(1 31 13 49)(2 32 14 50)(3 25 15 51)(4 26 16 52)(5 27 9 53)(6 28 10 54)(7 29 11 55)(8 30 12 56)(17 62 48 39)(18 63 41 40)(19 64 42 33)(20 57 43 34)(21 58 44 35)(22 59 45 36)(23 60 46 37)(24 61 47 38)
(1 38 13 61)(2 17 14 48)(3 63 15 40)(4 42 16 19)(5 34 9 57)(6 21 10 44)(7 59 11 36)(8 46 12 23)(18 51 41 25)(20 27 43 53)(22 55 45 29)(24 31 47 49)(26 64 52 33)(28 35 54 58)(30 60 56 37)(32 39 50 62)
(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)
(1 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)(17 48)(18 45)(19 42)(20 47)(21 44)(22 41)(23 46)(24 43)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)(33 64)(34 61)(35 58)(36 63)(37 60)(38 57)(39 62)(40 59)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4J 4K ··· 4T 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 M4(2) C4≀C2 C8.C22 kernel Q8⋊5M4(2) Q8⋊C8 C4×M4(2) C42.6C4 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 C42 C22×C4 Q8 C22 C4 # reps 1 4 1 1 1 2 4 2 2 2 8 8 2

Matrix representation of Q85M4(2) in GL4(𝔽17) generated by

 16 2 0 0 16 1 0 0 0 0 1 0 0 0 0 1
,
 0 7 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 5 12 0 0 11 0 0 0 0 0 8 15 0 0 13 9
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 9 1
`G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,7,0,0,0,0,0,1,0,0,0,0,1],[5,11,0,0,12,0,0,0,0,0,8,13,0,0,15,9],[16,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1] >;`

Q85M4(2) in GAP, Magma, Sage, TeX

`Q_8\rtimes_5M_4(2)`
`% in TeX`

`G:=Group("Q8:5M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,223);`
`// by ID`

`G=gap.SmallGroup(128,223);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,184,1123,570,136,172]);`
`// Polycyclic`

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

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