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## G = D4○(C22⋊C8)  order 128 = 27

### Central product of D4 and C22⋊C8

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4○(C22⋊C8)
G = < a,b,c,d,e | a4=b2=c2=d2=1, e4=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 636 in 387 conjugacy classes, 174 normal (16 characteristic)
C1, C2 [×3], C2 [×11], C4 [×8], C4 [×4], C22, C22 [×8], C22 [×29], C8 [×8], C2×C4 [×2], C2×C4 [×18], C2×C4 [×26], D4 [×12], D4 [×18], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×15], C2×C8 [×8], C2×C8 [×12], M4(2) [×12], C22×C4, C22×C4 [×15], C22×C4 [×12], C2×D4 [×12], C2×D4 [×12], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×8], C4○D4 [×28], C24 [×3], C22⋊C8, C22⋊C8 [×15], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×8], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×6], C2×C4○D4 [×8], C2×C22⋊C8 [×3], C24.4C4 [×3], (C22×C8)⋊C2 [×6], C2×C8○D4 [×2], C22×C4○D4, D4○(C22⋊C8)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C8○D4 [×2], C23×C4, C22×D4 [×2], C22×C22⋊C4, C2×C8○D4, Q8○M4(2), D4○(C22⋊C8)

Smallest permutation representation of D4○(C22⋊C8)
On 32 points
Generators in S32
```(1 29 5 25)(2 30 6 26)(3 31 7 27)(4 32 8 28)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 29)(27 31)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 ··· 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 4 1 1 1 1 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C8○D4 Q8○M4(2) kernel D4○(C22⋊C8) C2×C22⋊C8 C24.4C4 (C22×C8)⋊C2 C2×C8○D4 C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C4○D4 C22 C2 # reps 1 3 3 6 2 1 6 2 8 8 8 2

Matrix representation of D4○(C22⋊C8) in GL4(𝔽17) generated by

 13 0 0 0 0 4 0 0 0 0 16 0 0 0 0 16
,
 0 4 0 0 13 0 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 8 16
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 2 0 0 0 0 2 0 0 0 0 13 1 0 0 2 4
`G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,8,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,2,0,0,0,0,13,2,0,0,1,4] >;`

D4○(C22⋊C8) in GAP, Magma, Sage, TeX

`D_4\circ (C_2^2\rtimes C_8)`
`% in TeX`

`G:=Group("D4o(C2^2:C8)");`
`// GroupNames label`

`G:=SmallGroup(128,1612);`
`// by ID`

`G=gap.SmallGroup(128,1612);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,124]);`
`// Polycyclic`

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

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