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## G = C8.C4order 32 = 25

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

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

Aliases: C8.1C4, C22.Q8, C4.19D4, M4(2).2C2, C4.8(C2×C4), (C2×C8).5C2, C2.5(C4⋊C4), (C2×C4).19C22, SmallGroup(32,15)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8.C4
G = < a,b | a8=1, b4=a4, bab-1=a-1 >

Character table of C8.C4

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 1 1 2 2 2 2 2 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 -1 -1 1 -1 1 -1 1 -i -i i i linear of order 4 ρ6 1 1 -1 -1 -1 1 1 -1 1 -1 i -i -i i linear of order 4 ρ7 1 1 -1 -1 -1 1 1 -1 1 -1 -i i i -i linear of order 4 ρ8 1 1 -1 -1 -1 1 -1 1 -1 1 i i -i -i linear of order 4 ρ9 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 2i -2i 0 -√2 √-2 √2 -√-2 0 0 0 0 complex faithful ρ12 2 -2 0 2i -2i 0 √2 -√-2 -√2 √-2 0 0 0 0 complex faithful ρ13 2 -2 0 -2i 2i 0 -√2 -√-2 √2 √-2 0 0 0 0 complex faithful ρ14 2 -2 0 -2i 2i 0 √2 √-2 -√2 -√-2 0 0 0 0 complex faithful

Permutation representations of C8.C4
On 16 points - transitive group 16T49
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9 3 15 5 13 7 11)(2 16 4 14 6 12 8 10)```

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

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

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

`G:=TransitiveGroup(16,49);`

C8.C4 is a maximal subgroup of
C8.Q8  C8.4Q8  M4(2).C4  C40.C4  D10.Q8  C4.19S3≀C2  C62.2Q8  (C3×C24).C4  C8.(C32⋊C4)  C104.C4  C104.1C4
C4p.D4: D8.C4  M5(2)⋊C2  C8.17D4  C8○D8  C8.26D4  D4.3D4  D4.4D4  D4.5D4 ...
C8.C4 is a maximal quotient of
C4.C42  C40.C4  D10.Q8  C4.19S3≀C2  C62.2Q8  (C3×C24).C4  C8.(C32⋊C4)  C104.C4  C104.1C4
C4.D4p: C81C8  C24.C4  C40.6C4  C56.C4  C88.C4  C104.6C4 ...
(C2×C2p).Q8: C82C8  C12.53D4  C20.53D4  C28.53D4  C44.53D4  C52.53D4 ...

Matrix representation of C8.C4 in GL2(𝔽17) generated by

 9 0 7 2
,
 2 15 4 15
`G:=sub<GL(2,GF(17))| [9,7,0,2],[2,4,15,15] >;`

C8.C4 in GAP, Magma, Sage, TeX

`C_8.C_4`
`% in TeX`

`G:=Group("C8.C4");`
`// GroupNames label`

`G:=SmallGroup(32,15);`
`// by ID`

`G=gap.SmallGroup(32,15);`
`# by ID`

`G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,302,72,58]);`
`// Polycyclic`

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

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