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## G = M4(2)⋊8D4order 128 = 27

### 2nd semidirect product of M4(2) and D4 acting via D4/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)⋊8D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊8D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊8D4
 Upper central C1 — C22 — C2×C42 — M4(2)⋊8D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊8D4

Generators and relations for M4(2)⋊8D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 516 in 276 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×12], Q8 [×16], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], M4(2) [×8], SD16 [×16], Q16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×6], C2×Q8 [×10], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×8], C2×Q16 [×8], C8.C22 [×16], C22×Q8 [×2], C2×C4○D4 [×2], C4×M4(2), C85D4 [×2], C4⋊Q16 [×2], C8.2D4 [×4], C2×C4⋊Q8, C22.26C24, C2×C8.C22 [×4], M4(2)⋊8D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C8.C22 [×4], C22×D4 [×3], C2×C41D4, C2×C8.C22 [×2], M4(2)⋊8D4

Smallest permutation representation of M4(2)⋊8D4
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 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)(49 62)(50 59)(51 64)(52 61)(53 58)(54 63)(55 60)(56 57)
(1 11 64 34)(2 12 57 35)(3 13 58 36)(4 14 59 37)(5 15 60 38)(6 16 61 39)(7 9 62 40)(8 10 63 33)(17 52 44 27)(18 53 45 28)(19 54 46 29)(20 55 47 30)(21 56 48 31)(22 49 41 32)(23 50 42 25)(24 51 43 26)
(1 64)(2 59)(3 62)(4 57)(5 60)(6 63)(7 58)(8 61)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 52)(26 55)(27 50)(28 53)(29 56)(30 51)(31 54)(32 49)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C8.C22 kernel M4(2)⋊8D4 C4×M4(2) C8⋊5D4 C4⋊Q16 C8.2D4 C2×C4⋊Q8 C22.26C24 C2×C8.C22 C42 M4(2) C22×C4 C4 # reps 1 1 2 2 4 1 1 4 2 8 2 4

Matrix representation of M4(2)⋊8D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 7 7 9 9 0 0 10 16 0 8 0 0 1 10 0 6 0 0 16 6 10 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 16 15 0 0 1 0 0 0 0 0 0 0 0 16
,
 1 15 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 16 1 1

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,10,1,16,0,0,7,16,10,6,0,0,9,0,0,10,0,0,9,8,6,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,0,0,15,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,1] >;`

M4(2)⋊8D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_8D_4`
`% in TeX`

`G:=Group("M4(2):8D4");`
`// GroupNames label`

`G:=SmallGroup(128,1884);`
`// by ID`

`G=gap.SmallGroup(128,1884);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,184,1018,2804,172]);`
`// Polycyclic`

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

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