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G = C8.C8order 64 = 26

1st non-split extension by C8 of C8 acting via C8/C4=C2

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

Aliases: C8.1C8, C8.6Q8, C8.29D4, C42.8C4, M5(2).4C2, C22.5M4(2), (C2×C8).9C4, C4.8(C2×C8), C2.5(C4⋊C8), (C4×C8).10C2, C4.20(C4⋊C4), (C2×C8).96C22, (C2×C4).68(C2×C4), SmallGroup(64,45)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8.C8
G = < a,b | a8=1, b8=a4, bab-1=a3 >

Character table of C8.C8

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 1 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 4 4 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 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 -1 -1 -1 -1 -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 -1 -1 -1 -1 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 -1 1 1 -1 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 -1 1 1 -1 -i -i i -i i i -i i linear of order 4 ρ9 1 1 -1 -1 -1 -i 1 i i -i i i -i -i 1 -1 -i -1 1 i ζ83 ζ87 ζ8 ζ87 ζ85 ζ85 ζ83 ζ8 linear of order 8 ρ10 1 1 -1 -1 -1 -i 1 i i -i i i -i -i 1 -1 -i -1 1 i ζ87 ζ83 ζ85 ζ83 ζ8 ζ8 ζ87 ζ85 linear of order 8 ρ11 1 1 -1 -1 -1 -i 1 i i -i -i -i i i -1 1 i 1 -1 -i ζ8 ζ85 ζ87 ζ8 ζ83 ζ87 ζ85 ζ83 linear of order 8 ρ12 1 1 -1 -1 -1 -i 1 i i -i -i -i i i -1 1 i 1 -1 -i ζ85 ζ8 ζ83 ζ85 ζ87 ζ83 ζ8 ζ87 linear of order 8 ρ13 1 1 -1 -1 -1 i 1 -i -i i -i -i i i 1 -1 i -1 1 -i ζ85 ζ8 ζ87 ζ8 ζ83 ζ83 ζ85 ζ87 linear of order 8 ρ14 1 1 -1 -1 -1 i 1 -i -i i i i -i -i -1 1 -i 1 -1 i ζ87 ζ83 ζ8 ζ87 ζ85 ζ8 ζ83 ζ85 linear of order 8 ρ15 1 1 -1 -1 -1 i 1 -i -i i -i -i i i 1 -1 i -1 1 -i ζ8 ζ85 ζ83 ζ85 ζ87 ζ87 ζ8 ζ83 linear of order 8 ρ16 1 1 -1 -1 -1 i 1 -i -i i i i -i -i -1 1 -i 1 -1 i ζ83 ζ87 ζ85 ζ83 ζ8 ζ85 ζ87 ζ8 linear of order 8 ρ17 2 2 -2 2 2 0 -2 0 0 0 2 2 2 2 0 0 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 2 0 -2 0 0 0 -2 -2 -2 -2 0 0 2 0 0 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 2 2 -2 -2 0 -2 0 0 0 2i 2i -2i -2i 0 0 2i 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 2 -2 -2 0 -2 0 0 0 -2i -2i 2i 2i 0 0 -2i 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 2 -2 0 2i -2i 1+i 0 1-i -1+i -1-i 2ζ83 2ζ87 2ζ85 2ζ8 -√2 -√-2 0 √-2 √2 0 0 0 0 0 0 0 0 0 complex faithful ρ22 2 -2 0 2i -2i 1+i 0 1-i -1+i -1-i 2ζ87 2ζ83 2ζ8 2ζ85 √2 √-2 0 -√-2 -√2 0 0 0 0 0 0 0 0 0 complex faithful ρ23 2 -2 0 -2i 2i -1+i 0 -1-i 1+i 1-i 2ζ8 2ζ85 2ζ87 2ζ83 -√2 √-2 0 -√-2 √2 0 0 0 0 0 0 0 0 0 complex faithful ρ24 2 -2 0 -2i 2i 1-i 0 1+i -1-i -1+i 2ζ85 2ζ8 2ζ83 2ζ87 -√2 √-2 0 -√-2 √2 0 0 0 0 0 0 0 0 0 complex faithful ρ25 2 -2 0 2i -2i -1-i 0 -1+i 1-i 1+i 2ζ87 2ζ83 2ζ8 2ζ85 -√2 -√-2 0 √-2 √2 0 0 0 0 0 0 0 0 0 complex faithful ρ26 2 -2 0 2i -2i -1-i 0 -1+i 1-i 1+i 2ζ83 2ζ87 2ζ85 2ζ8 √2 √-2 0 -√-2 -√2 0 0 0 0 0 0 0 0 0 complex faithful ρ27 2 -2 0 -2i 2i -1+i 0 -1-i 1+i 1-i 2ζ85 2ζ8 2ζ83 2ζ87 √2 -√-2 0 √-2 -√2 0 0 0 0 0 0 0 0 0 complex faithful ρ28 2 -2 0 -2i 2i 1-i 0 1+i -1-i -1+i 2ζ8 2ζ85 2ζ87 2ζ83 √2 -√-2 0 √-2 -√2 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C8.C8
On 16 points - transitive group 16T124
Generators in S16
```(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,124);`

C8.C8 is a maximal subgroup of
C8≀C2  C8.32D8  C8.24D8  C8.25D8  C8.29D8  C8.1Q16  M4(2).1C8  C8.5M4(2)  C8.19M4(2)  C8.3D8  D83D4  C8.5D8  D83Q8  D8.2Q8  C42.9F5
C8p.C8: C16.C8  C16.3C8  C24.1C8  C40.7C8  C40.1C8  C56.16Q8 ...
C8p.D4: C16○D8  D8.C8  C24.97D4  C40.9Q8  C56.9Q8 ...
C8.C8 is a maximal quotient of
C82C16
C8.D4p: C8.36D8  C24.1C8  C40.7C8  C56.16Q8 ...
C2p.(C4⋊C8): C42.7C8  C24.97D4  C40.9Q8  C42.9F5  C40.1C8  C56.9Q8 ...

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

 9 0 0 15
,
 0 1 2 0
`G:=sub<GL(2,GF(17))| [9,0,0,15],[0,2,1,0] >;`

C8.C8 in GAP, Magma, Sage, TeX

`C_8.C_8`
`% in TeX`

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

`G:=SmallGroup(64,45);`
`// by ID`

`G=gap.SmallGroup(64,45);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,650,158,69,88]);`
`// Polycyclic`

`G:=Group<a,b|a^8=1,b^8=a^4,b*a*b^-1=a^3>;`
`// generators/relations`

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