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## G = D4⋊4M4(2)  order 128 = 27

### 2nd semidirect product of D4 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 — D4⋊4M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — D4⋊4M4(2)
 Lower central C1 — C2 — C2×C4 — D4⋊4M4(2)
 Upper central C1 — C2×C4 — C2×C42 — D4⋊4M4(2)
 Jennings C1 — C22 — C22 — C42 — D4⋊4M4(2)

Generators and relations for D44M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd=a2b, dcd=c5 >

Subgroups: 244 in 133 conjugacy classes, 52 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×2], C4 [×7], C22, C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×17], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×6], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, D4⋊C8 [×2], Q8⋊C8 [×2], C4×M4(2), C4⋊M4(2), C4×C4○D4, D44M4(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, C8.C22, C24.4C4, C23.36D4, C2×C4≀C2, D44M4(2)

Smallest permutation representation of D44M4(2)
On 64 points
Generators in S64
```(1 52 59 17)(2 18 60 53)(3 54 61 19)(4 20 62 55)(5 56 63 21)(6 22 64 49)(7 50 57 23)(8 24 58 51)(9 45 34 31)(10 32 35 46)(11 47 36 25)(12 26 37 48)(13 41 38 27)(14 28 39 42)(15 43 40 29)(16 30 33 44)
(1 29 59 43)(2 33 60 16)(3 31 61 45)(4 35 62 10)(5 25 63 47)(6 37 64 12)(7 27 57 41)(8 39 58 14)(9 54 34 19)(11 56 36 21)(13 50 38 23)(15 52 40 17)(18 30 53 44)(20 32 55 46)(22 26 49 48)(24 28 51 42)
(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 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 54)(10 51)(11 56)(12 53)(13 50)(14 55)(15 52)(16 49)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(41 57)(42 62)(43 59)(44 64)(45 61)(46 58)(47 63)(48 60)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4L 4M ··· 4S 8A ··· 8H 8I 8J 8K 8L order 1 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 4 4 4 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 1 1 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 M4(2) M4(2) C4≀C2 C8⋊C22 C8.C22 kernel D4⋊4M4(2) D4⋊C8 Q8⋊C8 C4×M4(2) C4⋊M4(2) C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C42 C22×C4 D4 Q8 C4 C4 C4 # reps 1 2 2 1 1 1 2 2 2 2 2 2 4 4 8 1 1

Matrix representation of D44M4(2) in GL4(𝔽17) generated by

 0 16 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 0 4 0 0 4 0 0 0 0 0 16 0 0 0 0 16
,
 6 6 0 0 6 11 0 0 0 0 10 15 0 0 14 7
,
 0 13 0 0 4 0 0 0 0 0 16 0 0 0 7 1
`G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,16,0,0,0,0,16],[6,6,0,0,6,11,0,0,0,0,10,14,0,0,15,7],[0,4,0,0,13,0,0,0,0,0,16,7,0,0,0,1] >;`

D44M4(2) in GAP, Magma, Sage, TeX

`D_4\rtimes_4M_4(2)`
`% in TeX`

`G:=Group("D4:4M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,221);`
`// by ID`

`G=gap.SmallGroup(128,221);`
`# by ID`

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

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

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