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## G = C4.4D8order 64 = 26

### 4th non-split extension by C4 of D8 acting via D8/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.4D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4.4D8
 Lower central C1 — C2 — C2×C4 — C4.4D8
 Upper central C1 — C22 — C42 — C4.4D8
 Jennings C1 — C2 — C2 — C2×C4 — C4.4D8

Generators and relations for C4.4D8
G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 129 in 59 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C4×C8, D4⋊C4, C41D4, C4⋊Q8, C4.4D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4.4D4, C2×D8, C2×SD16, C4.4D8

Character table of C4.4D8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 8 2 2 2 2 2 2 8 8 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ7 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 0 0 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 0 0 2 0 -2 0 0 -√2 √2 -√2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ12 2 -2 -2 2 0 0 0 0 0 -2 0 2 0 0 -√2 -√2 -√2 √2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 0 0 0 2 0 -2 0 0 √2 -√2 √2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ14 2 -2 -2 2 0 0 0 0 0 -2 0 2 0 0 √2 √2 √2 -√2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ15 2 -2 2 -2 0 0 0 0 -2 0 2 0 0 0 2i 0 -2i 0 -2i 0 2i 0 complex lifted from C4○D4 ρ16 2 2 -2 -2 0 0 -2 2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 -2 2 -2 0 0 0 0 2 0 -2 0 0 0 0 -2i 0 -2i 0 2i 0 2i complex lifted from C4○D4 ρ18 2 -2 2 -2 0 0 0 0 -2 0 2 0 0 0 -2i 0 2i 0 2i 0 -2i 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 0 0 2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ20 2 -2 2 -2 0 0 0 0 2 0 -2 0 0 0 0 2i 0 2i 0 -2i 0 -2i complex lifted from C4○D4 ρ21 2 2 -2 -2 0 0 -2 2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ22 2 2 -2 -2 0 0 2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16

Smallest permutation representation of C4.4D8
On 32 points
Generators in S32
```(1 21 27 10)(2 22 28 11)(3 23 29 12)(4 24 30 13)(5 17 31 14)(6 18 32 15)(7 19 25 16)(8 20 26 9)
(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 26 27 8)(2 7 28 25)(3 32 29 6)(4 5 30 31)(9 21 20 10)(11 19 22 16)(12 15 23 18)(13 17 24 14)```

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

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

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

Matrix representation of C4.4D8 in GL4(𝔽17) generated by

 16 2 0 0 16 1 0 0 0 0 0 13 0 0 13 0
,
 7 10 0 0 12 0 0 0 0 0 0 1 0 0 1 0
,
 7 10 0 0 12 10 0 0 0 0 0 1 0 0 16 0
`G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,0,13,0,0,13,0],[7,12,0,0,10,0,0,0,0,0,0,1,0,0,1,0],[7,12,0,0,10,10,0,0,0,0,0,16,0,0,1,0] >;`

C4.4D8 in GAP, Magma, Sage, TeX

`C_4._4D_8`
`% in TeX`

`G:=Group("C4.4D8");`
`// GroupNames label`

`G:=SmallGroup(64,167);`
`// by ID`

`G=gap.SmallGroup(64,167);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,50,963,117,1444,88]);`
`// Polycyclic`

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

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