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## G = D4.C8order 64 = 26

### The non-split extension by D4 of C8 acting via C8/C4=C2

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

Aliases: D4.C8, Q8.C8, C8.25D4, M5(2)⋊4C2, M4(2).3C4, C22.1M4(2), (C2×C16)⋊2C2, C4.3(C2×C8), C8○D4.2C2, C4○D4.1C4, C2.8(C22⋊C8), (C2×C8).95C22, C4.30(C22⋊C4), (C2×C4).40(C2×C4), SmallGroup(64,31)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4.C8
G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, cbc-1=ab >

Character table of D4.C8

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D 16E 16F 16G 16H 16I 16J 16K 16L size 1 1 2 4 1 1 2 4 1 1 1 1 2 2 4 4 2 2 2 2 2 2 2 2 4 4 4 4 ρ1 1 1 1 1 1 1 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 -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 -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 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 i i -i -i i -i -i i i i -i -i linear of order 4 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -i -i i i -i i i -i -i -i i i linear of order 4 ρ7 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 i i -i -i i -i -i i -i -i i i linear of order 4 ρ8 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -i -i i i -i i i -i i i -i -i linear of order 4 ρ9 1 1 -1 -1 -1 -1 1 1 i -i i -i -i i -i i ζ87 ζ87 ζ8 ζ8 ζ83 ζ85 ζ85 ζ83 ζ87 ζ83 ζ8 ζ85 linear of order 8 ρ10 1 1 -1 1 -1 -1 1 -1 -i i -i i i -i -i i ζ85 ζ85 ζ83 ζ83 ζ8 ζ87 ζ87 ζ8 ζ8 ζ85 ζ87 ζ83 linear of order 8 ρ11 1 1 -1 1 -1 -1 1 -1 -i i -i i i -i -i i ζ8 ζ8 ζ87 ζ87 ζ85 ζ83 ζ83 ζ85 ζ85 ζ8 ζ83 ζ87 linear of order 8 ρ12 1 1 -1 -1 -1 -1 1 1 i -i i -i -i i -i i ζ83 ζ83 ζ85 ζ85 ζ87 ζ8 ζ8 ζ87 ζ83 ζ87 ζ85 ζ8 linear of order 8 ρ13 1 1 -1 1 -1 -1 1 -1 i -i i -i -i i i -i ζ83 ζ83 ζ85 ζ85 ζ87 ζ8 ζ8 ζ87 ζ87 ζ83 ζ8 ζ85 linear of order 8 ρ14 1 1 -1 -1 -1 -1 1 1 -i i -i i i -i i -i ζ85 ζ85 ζ83 ζ83 ζ8 ζ87 ζ87 ζ8 ζ85 ζ8 ζ83 ζ87 linear of order 8 ρ15 1 1 -1 1 -1 -1 1 -1 i -i i -i -i i i -i ζ87 ζ87 ζ8 ζ8 ζ83 ζ85 ζ85 ζ83 ζ83 ζ87 ζ85 ζ8 linear of order 8 ρ16 1 1 -1 -1 -1 -1 1 1 -i i -i i i -i i -i ζ8 ζ8 ζ87 ζ87 ζ85 ζ83 ζ83 ζ85 ζ8 ζ85 ζ87 ζ83 linear of order 8 ρ17 2 2 -2 0 2 2 -2 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 2 2 -2 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 0 -2 -2 -2 0 -2i 2i -2i 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 2 0 -2 -2 -2 0 2i -2i 2i -2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 2 -2 0 0 -2i 2i 0 0 2ζ1614 2ζ162 2ζ166 2ζ1610 0 0 0 0 ζ1615+ζ1611 ζ167+ζ163 ζ1613+ζ169 ζ165+ζ16 ζ1611+ζ167 ζ169+ζ165 ζ1613+ζ16 ζ1615+ζ163 0 0 0 0 complex faithful ρ22 2 -2 0 0 2i -2i 0 0 2ζ162 2ζ1614 2ζ1610 2ζ166 0 0 0 0 ζ1613+ζ169 ζ165+ζ16 ζ1615+ζ1611 ζ167+ζ163 ζ1613+ζ16 ζ1615+ζ163 ζ1611+ζ167 ζ169+ζ165 0 0 0 0 complex faithful ρ23 2 -2 0 0 -2i 2i 0 0 2ζ1614 2ζ162 2ζ166 2ζ1610 0 0 0 0 ζ167+ζ163 ζ1615+ζ1611 ζ165+ζ16 ζ1613+ζ169 ζ1615+ζ163 ζ1613+ζ16 ζ169+ζ165 ζ1611+ζ167 0 0 0 0 complex faithful ρ24 2 -2 0 0 2i -2i 0 0 2ζ1610 2ζ166 2ζ162 2ζ1614 0 0 0 0 ζ169+ζ165 ζ1613+ζ16 ζ1615+ζ163 ζ1611+ζ167 ζ1613+ζ169 ζ167+ζ163 ζ1615+ζ1611 ζ165+ζ16 0 0 0 0 complex faithful ρ25 2 -2 0 0 -2i 2i 0 0 2ζ166 2ζ1610 2ζ1614 2ζ162 0 0 0 0 ζ1611+ζ167 ζ1615+ζ163 ζ1613+ζ16 ζ169+ζ165 ζ167+ζ163 ζ1613+ζ169 ζ165+ζ16 ζ1615+ζ1611 0 0 0 0 complex faithful ρ26 2 -2 0 0 2i -2i 0 0 2ζ1610 2ζ166 2ζ162 2ζ1614 0 0 0 0 ζ1613+ζ16 ζ169+ζ165 ζ1611+ζ167 ζ1615+ζ163 ζ165+ζ16 ζ1615+ζ1611 ζ167+ζ163 ζ1613+ζ169 0 0 0 0 complex faithful ρ27 2 -2 0 0 2i -2i 0 0 2ζ162 2ζ1614 2ζ1610 2ζ166 0 0 0 0 ζ165+ζ16 ζ1613+ζ169 ζ167+ζ163 ζ1615+ζ1611 ζ169+ζ165 ζ1611+ζ167 ζ1615+ζ163 ζ1613+ζ16 0 0 0 0 complex faithful ρ28 2 -2 0 0 -2i 2i 0 0 2ζ166 2ζ1610 2ζ1614 2ζ162 0 0 0 0 ζ1615+ζ163 ζ1611+ζ167 ζ169+ζ165 ζ1613+ζ16 ζ1615+ζ1611 ζ165+ζ16 ζ1613+ζ169 ζ167+ζ163 0 0 0 0 complex faithful

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

D4.C8 is a maximal subgroup of
Q16.10D4  Q16.D4  D8.3D4  D8.12D4  C8.7S4
D4p.C8: C16○D8  D8.C8  D12.C8  Dic6.C8  D20.3C8  D20.4C8  D20.C8  D28.C8 ...
(Cp×D4).C8: M5(2)⋊12C22  C24.99D4  C40.92D4  D4.(C5⋊C8)  C56.92D4 ...
D4.C8 is a maximal quotient of
C23.7M4(2)  Q8⋊C16  C8.17Q16  M5(2)⋊7C4  D20.C8
C8.D4p: D4⋊C16  C8.31D8  D12.C8  Dic6.C8  D20.3C8  D20.4C8  D28.C8  Dic14.C8 ...
(Cp×D4).C8: C23.M4(2)  C24.99D4  C40.92D4  D4.(C5⋊C8)  C56.92D4 ...

Matrix representation of D4.C8 in GL2(𝔽17) generated by

 0 9 15 0
,
 0 9 2 0
,
 16 8 2 16
`G:=sub<GL(2,GF(17))| [0,15,9,0],[0,2,9,0],[16,2,8,16] >;`

D4.C8 in GAP, Magma, Sage, TeX

`D_4.C_8`
`% in TeX`

`G:=Group("D4.C8");`
`// GroupNames label`

`G:=SmallGroup(64,31);`
`// by ID`

`G=gap.SmallGroup(64,31);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,332,158,489,69,88]);`
`// Polycyclic`

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

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