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## G = C8.C22order 32 = 25

### The non-split extension by C8 of C22 acting faithfully

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

Aliases: C8.C22, Q162C2, C4.15D4, SD162C2, C4.6C23, C22.6D4, M4(2)⋊2C2, D4.3C22, Q8.3C22, (C2×Q8)⋊4C2, C4○D4.2C2, C2.16(C2×D4), (C2×C4).7C22, 2-Sylow(ASigmaL(2,9)), SmallGroup(32,44)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8.C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×Q8 — C8.C22
 Lower central C1 — C2 — C4 — C8.C22
 Upper central C1 — C2 — C2×C4 — C8.C22
 Jennings C1 — C2 — C2 — C4 — C8.C22

Generators and relations for C8.C22
G = < a,b,c | a8=b2=c2=1, bab=a3, cac=a5, cbc=a4b >

Character table of C8.C22

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B size 1 1 2 4 2 2 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 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 linear of order 2 ρ4 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 linear of order 2 ρ6 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 linear of order 2 ρ8 1 1 1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ9 2 2 2 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 -4 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C8.C22
On 16 points - transitive group 16T32
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)```

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

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

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

`G:=TransitiveGroup(16,32);`

On 16 points - transitive group 16T50
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 15)(3 10)(4 13)(5 16)(6 11)(7 14)(8 9)
(2 6)(4 8)(10 14)(12 16)```

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

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

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

`G:=TransitiveGroup(16,50);`

C8.C22 is a maximal subgroup of
Q8.D6  C4.S4  C32⋊Q16⋊C2  C62.15D4
D4.D2p: D4.9D4  D4.10D4  D4.3D4  D4.5D4  D4○SD16  Q8○D8  D4.D6  Q8.14D6 ...
C4p.C23: D8⋊C22  C8.D6  Q16⋊S3  Q8.11D6  C8.D10  Q16⋊D5  C20.C23  C8.D14 ...
C8.C22 is a maximal quotient of
C23.36D4  C23.38D4  M4(2)⋊C4  SD16⋊C4  Q16⋊C4  Q8⋊D4  C8⋊D4  D4⋊Q8  Q8⋊Q8  Q8.Q8  C22.D8  C23.47D4  C23.20D4  C42.28C22  C42.30C22  C8⋊Q8  C32⋊Q16⋊C2  C62.15D4
D4.D2p: D4.7D4  D4.D4  D4.D6  Q8.14D6  SD16⋊D5  D4.9D10  SD16⋊D7  D4.9D14 ...
C8.D2p: C8.D4  C8.2D4  C8.D6  Q16⋊S3  C8.D10  Q16⋊D5  C8.D14  Q16⋊D7 ...
Q8.D2p: C22⋊Q16  C42Q16  Q8.D4  Q8.11D6  C20.C23  C28.C23  C44.C23  Q8.D26 ...

Matrix representation of C8.C22 in GL4(𝔽3) generated by

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

C8.C22 in GAP, Magma, Sage, TeX

`C_8.C_2^2`
`% in TeX`

`G:=Group("C8.C2^2");`
`// GroupNames label`

`G:=SmallGroup(32,44);`
`// by ID`

`G=gap.SmallGroup(32,44);`
`# by ID`

`G:=PCGroup([5,-2,2,2,-2,-2,101,86,302,483,248,58]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^2=c^2=1,b*a*b=a^3,c*a*c=a^5,c*b*c=a^4*b>;`
`// generators/relations`

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