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## G = M4(2)⋊17D4order 128 = 27

### 4th semidirect product of M4(2) and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊17D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2)⋊17D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊17D4
 Upper central C1 — C22 — C2×C4○D4 — M4(2)⋊17D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊17D4

Generators and relations for M4(2)⋊17D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a-1, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 436 in 236 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×11], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×10], C23, C23 [×2], C23, C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], M4(2) [×4], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×6], C22⋊Q8, C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×4], C22×Q8, C2×C4○D4 [×2], C2×Q8⋊C4, C23.24D4, M4(2)⋊C4, C88D4 [×2], C8.18D4 [×2], C8⋊D4 [×2], C8.D4 [×2], C23.38C23, C22.31C24, C2×C8○D4, C2×C8.C22, M4(2)⋊17D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16, Q8○D8, M4(2)⋊17D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G ··· 4M 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 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 4 4 8 2 2 2 2 4 4 8 ··· 8 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○SD16 Q8○D8 kernel M4(2)⋊17D4 C2×Q8⋊C4 C23.24D4 M4(2)⋊C4 C8⋊8D4 C8.18D4 C8⋊D4 C8.D4 C23.38C23 C22.31C24 C2×C8○D4 C2×C8.C22 M4(2) C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 4 3 1 4 2 2

Matrix representation of M4(2)⋊17D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 3 4 13 0 0 14 3 4 4 0 0 13 4 14 14 0 0 13 13 3 14
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 2 0 0 0 0 16 16 0 0 0 0 0 0 11 0 0 4 0 0 0 6 4 0 0 0 0 4 11 0 0 0 4 0 0 6
,
 1 0 0 0 0 0 16 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 0 0 1

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

M4(2)⋊17D4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_{17}D_4`
`% in TeX`

`G:=Group("M4(2):17D4");`
`// GroupNames label`

`G:=SmallGroup(128,1795);`
`// by ID`

`G=gap.SmallGroup(128,1795);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,521,2804,172]);`
`// Polycyclic`

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