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## G = C8.26D4order 64 = 26

### 13rd non-split extension by C8 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8.26D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — C8.26D4
 Lower central C1 — C2 — C4 — C8.26D4
 Upper central C1 — C4 — C2×C8 — C8.26D4
 Jennings C1 — C2 — C2 — C2×C4 — C8.26D4

Generators and relations for C8.26D4
G = < a,b,c | a8=b4=1, c2=a2, bab-1=cac-1=a5, cbc-1=a2b-1 >

Character table of C8.26D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 2 4 4 1 1 2 4 4 4 4 2 2 2 2 4 4 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 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 1 1 -1 1 -1 -1 -1 1 -i -1 1 i -i i i -i -1 -i -i 1 i i linear of order 4 ρ10 1 1 -1 1 1 -1 -1 1 i -1 -1 -i -i i i -i 1 i -i -1 i -i linear of order 4 ρ11 1 1 -1 -1 1 -1 -1 1 -i 1 -1 i -i i i -i -1 i i 1 -i -i linear of order 4 ρ12 1 1 -1 -1 -1 -1 -1 1 i 1 1 -i -i i i -i 1 -i i -1 -i i linear of order 4 ρ13 1 1 -1 -1 -1 -1 -1 1 -i 1 1 i i -i -i i 1 i -i -1 i -i linear of order 4 ρ14 1 1 -1 -1 1 -1 -1 1 i 1 -1 -i i -i -i i -1 -i -i 1 i i linear of order 4 ρ15 1 1 -1 1 1 -1 -1 1 -i -1 -1 i i -i -i i 1 -i i -1 -i i linear of order 4 ρ16 1 1 -1 1 -1 -1 -1 1 i -1 1 -i i -i -i i -1 i i 1 -i -i linear of order 4 ρ17 2 2 -2 0 0 2 2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 0 2 2 -2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 0 0 -2 -2 -2 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 0 0 -2 -2 -2 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 4 -4 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C8.26D4
On 16 points - transitive group 16T113
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)
(1 15 3 9 5 11 7 13)(2 12 4 14 6 16 8 10)```

`G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15,3,9,5,11,7,13)(2,12,4,14,6,16,8,10)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15,3,9,5,11,7,13)(2,12,4,14,6,16,8,10) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12)], [(1,15,3,9,5,11,7,13),(2,12,4,14,6,16,8,10)]])`

`G:=TransitiveGroup(16,113);`

C8.26D4 is a maximal subgroup of
C42.283C23  D811D4  D8.13D4  C8.5S4
D8p⋊C4: D16⋊C4  D244C4  D2410C4  D4010C4  D4016C4  Q16⋊F5  D564C4  D5610C4 ...
M4(2).D2p: M4(2).51D4  M4(2)○D8  M4(2).22D6  C24.54D4  M4(2).22D10  C40.50D4  M4(2).22D14  C56.50D4 ...
C8p.(C2×C4): Q32⋊C4  D84Dic3  D84Dic5  D8⋊F5  SD162F5  D84Dic7 ...
C8.26D4 is a maximal quotient of
SD16⋊C8  Q165C8  D85C8  C89D8  C812SD16  C815SD16  C89Q16  C8⋊M4(2)  C8.M4(2)  C83M4(2)  D4.3C42  C8.5C42  M4(2).3Q8  C42.28Q8  SD162F5  Q16⋊F5
M4(2).D2p: M4(2).42D4  M4(2).43D4  M4(2).24D4  M4(2).22D6  D2410C4  C24.54D4  M4(2).22D10  D4016C4 ...
(Cp×D8)⋊C4: C42.116D4  D84Dic3  D84Dic5  D8⋊F5  D84Dic7 ...
C42.D2p: C42.107D4  D244C4  D4010C4  D564C4 ...

Matrix representation of C8.26D4 in GL4(𝔽5) generated by

 0 0 2 0 0 0 0 1 4 0 0 0 0 3 0 0
,
 2 0 0 0 0 1 0 0 0 0 3 0 0 0 0 4
,
 0 0 0 4 0 0 2 0 0 4 0 0 2 0 0 0
`G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,3,2,0,0,0,0,1,0,0],[2,0,0,0,0,1,0,0,0,0,3,0,0,0,0,4],[0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,0] >;`

C8.26D4 in GAP, Magma, Sage, TeX

`C_8._{26}D_4`
`% in TeX`

`G:=Group("C8.26D4");`
`// GroupNames label`

`G:=SmallGroup(64,125);`
`// by ID`

`G=gap.SmallGroup(64,125);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^4=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b^-1>;`
`// generators/relations`

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