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G = M4(2)⋊5Q8order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for M4(2)⋊5Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 324 in 190 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×12], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×10], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C4×M4(2), M4(2)⋊C4 [×4], C83Q8 [×2], C82Q8 [×2], C8⋊Q8 [×4], C2×C4⋊Q8, C23.37C23, M4(2)⋊5Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, C2×C8⋊C22, C2×C8.C22, M4(2)⋊5Q8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D4 C8⋊C22 C8.C22 kernel M4(2)⋊5Q8 C4×M4(2) M4(2)⋊C4 C8⋊3Q8 C8⋊2Q8 C8⋊Q8 C2×C4⋊Q8 C23.37C23 C42 M4(2) C22×C4 C4 C4 # reps 1 1 4 2 2 4 1 1 2 8 2 2 2

Matrix representation of M4(2)⋊5Q8 in GL6(𝔽17)

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

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

M4(2)⋊5Q8 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_5Q_8`
`% in TeX`

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

`G:=SmallGroup(128,1897);`
`// by ID`

`G=gap.SmallGroup(128,1897);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,521,4037,124]);`
`// Polycyclic`

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

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