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## G = C8○D4⋊C4order 128 = 27

### 1st semidirect product of C8○D4 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8○D4⋊C4
G = < a,b,c,d | a8=c2=d4=1, b2=a4, ab=ba, ac=ca, dad-1=a-1, cbc=a4b, dbd-1=a6c, dcd-1=a2b >

Subgroups: 228 in 128 conjugacy classes, 58 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4≀C2, C4.Q8, C2.D8, C8.C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, M4(2)⋊4C4, C42⋊C22, C23.25D4, C2×C8.C4, C2×C8○D4, C8○D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, C8○D4⋊C4

Smallest permutation representation of C8○D4⋊C4
On 32 points
Generators in S32
```(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 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C4 D4 Q8 Q8 D4 C4○D4 C4○D4 C8○D4⋊C4 kernel C8○D4⋊C4 M4(2)⋊4C4 C42⋊C22 C23.25D4 C2×C8.C4 C2×C8○D4 C8○D4 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C1 # reps 1 2 2 1 1 1 8 4 1 1 2 2 2 4

Matrix representation of C8○D4⋊C4 in GL4(𝔽17) generated by

 11 11 0 0 3 0 0 0 0 0 11 11 0 0 3 0
,
 0 0 13 0 0 0 0 13 13 0 0 0 0 13 0 0
,
 0 0 1 2 0 0 16 16 16 15 0 0 1 1 0 0
,
 0 10 0 0 5 0 0 0 0 0 0 6 0 0 3 0
`G:=sub<GL(4,GF(17))| [11,3,0,0,11,0,0,0,0,0,11,3,0,0,11,0],[0,0,13,0,0,0,0,13,13,0,0,0,0,13,0,0],[0,0,16,1,0,0,15,1,1,16,0,0,2,16,0,0],[0,5,0,0,10,0,0,0,0,0,0,3,0,0,6,0] >;`

C8○D4⋊C4 in GAP, Magma, Sage, TeX

`C_8\circ D_4\rtimes C_4`
`% in TeX`

`G:=Group("C8oD4:C4");`
`// GroupNames label`

`G:=SmallGroup(128,546);`
`// by ID`

`G=gap.SmallGroup(128,546);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,718,172,1027]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^2=d^4=1,b^2=a^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^4*b,d*b*d^-1=a^6*c,d*c*d^-1=a^2*b>;`
`// generators/relations`

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