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## G = D4⋊5M4(2)  order 128 = 27

### 3rd 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⋊5M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×D4 — D4⋊5M4(2)
 Lower central C1 — C2 — C2×C4 — D4⋊5M4(2)
 Upper central C1 — C2×C4 — C2×C42 — D4⋊5M4(2)
 Jennings C1 — C22 — C22 — C42 — D4⋊5M4(2)

Generators and relations for D45M4(2)
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >

Subgroups: 340 in 158 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×18], C8 [×6], C2×C4 [×6], C2×C4 [×18], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×9], C2×D4 [×2], C2×D4 [×5], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8, C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C2×M4(2), C23×C4, C22×D4, D4⋊C8 [×4], C4×M4(2), C42.6C4, C2×C4×D4, D45M4(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, C23.37D4, C2×C4≀C2, D45M4(2)

Smallest permutation representation of D45M4(2)
On 32 points
Generators in S32
```(1 12 32 19)(2 13 25 20)(3 14 26 21)(4 15 27 22)(5 16 28 23)(6 9 29 24)(7 10 30 17)(8 11 31 18)
(1 28)(2 24)(3 7)(4 11)(5 32)(6 20)(8 15)(9 25)(10 21)(12 16)(13 29)(14 17)(18 27)(19 23)(22 31)(26 30)
(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)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)```

`G:=sub<Sym(32)| (1,12,32,19)(2,13,25,20)(3,14,26,21)(4,15,27,22)(5,16,28,23)(6,9,29,24)(7,10,30,17)(8,11,31,18), (1,28)(2,24)(3,7)(4,11)(5,32)(6,20)(8,15)(9,25)(10,21)(12,16)(13,29)(14,17)(18,27)(19,23)(22,31)(26,30), (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), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23)>;`

`G:=Group( (1,12,32,19)(2,13,25,20)(3,14,26,21)(4,15,27,22)(5,16,28,23)(6,9,29,24)(7,10,30,17)(8,11,31,18), (1,28)(2,24)(3,7)(4,11)(5,32)(6,20)(8,15)(9,25)(10,21)(12,16)(13,29)(14,17)(18,27)(19,23)(22,31)(26,30), (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), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23) );`

`G=PermutationGroup([(1,12,32,19),(2,13,25,20),(3,14,26,21),(4,15,27,22),(5,16,28,23),(6,9,29,24),(7,10,30,17),(8,11,31,18)], [(1,28),(2,24),(3,7),(4,11),(5,32),(6,20),(8,15),(9,25),(10,21),(12,16),(13,29),(14,17),(18,27),(19,23),(22,31),(26,30)], [(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)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,20),(10,17),(11,22),(12,19),(13,24),(14,21),(15,18),(16,23)])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 8A ··· 8H 8I 8J 8K 8L order 1 2 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 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 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 M4(2) C4≀C2 C8⋊C22 kernel D4⋊5M4(2) D4⋊C8 C4×M4(2) C42.6C4 C2×C4×D4 C2×C4⋊C4 C4×D4 C22×D4 C42 C22×C4 D4 C22 C4 # reps 1 4 1 1 1 2 4 2 2 2 8 8 2

Matrix representation of D45M4(2) in GL4(𝔽17) generated by

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

D45M4(2) in GAP, Magma, Sage, TeX

`D_4\rtimes_5M_4(2)`
`% in TeX`

`G:=Group("D4:5M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,222);`
`// by ID`

`G=gap.SmallGroup(128,222);`
`# by ID`

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

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