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## G = C42.3C4order 64 = 26

### 3rd non-split extension by C42 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42.3C4, (C2×C4).4D4, C4⋊Q8.3C2, (C2×Q8).3C4, C4.10D4.C2, (C2×Q8).2C22, C2.11(C23⋊C4), C22.14(C22⋊C4), (C2×C4).4(C2×C4), SmallGroup(64,37)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.3C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — C42.3C4
 Lower central C1 — C2 — C22 — C2×C4 — C42.3C4
 Upper central C1 — C2 — C22 — C2×Q8 — C42.3C4
 Jennings C1 — C2 — C22 — C2×Q8 — C42.3C4

Generators and relations for C42.3C4
G = < a,b,c | a4=b4=1, c4=b2, ab=ba, cac-1=a-1b, cbc-1=a2b >

Character table of C42.3C4

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 2 4 4 4 4 4 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 i -i -i i linear of order 4 ρ7 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 i i -i -i linear of order 4 ρ9 2 2 2 0 0 2 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 -2 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 -4 0 2 -2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ13 4 -4 0 -2 2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C42.3C4
On 16 points - transitive group 16T139
Generators in S16
```(1 5)(2 16 6 12)(3 7)(4 14 8 10)(9 13)(11 15)
(1 11 5 15)(2 12 6 16)(3 9 7 13)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,139);`

C42.3C4 is a maximal subgroup of
(C2×D4).135D4  (C2×D4).137D4  C42.F5  (Q8×C10).C4
(C2×Q8).D2p: C42.4D4  (C4×C8).C4  (C2×Q8).D4  C8⋊C4.C4  C4⋊Q8.C4  C42.16D4  C42.17D4  Q8≀C2 ...
C42.3C4 is a maximal quotient of
(C2×C42).C4  C423C8  C2.7C2≀C4  C42.F5  (Q8×C10).C4
(C2×C4).D4p: (C2×C4).D8  (C2×C12).D4  (C2×Q8).D10  (C2×Q8).D14 ...
(C2×Q8).D2p: (C2×Q8).Q8  C42.3Dic3  C42.3Dic5  C42.3Dic7 ...

Matrix representation of C42.3C4 in GL4(𝔽3) generated by

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

C42.3C4 in GAP, Magma, Sage, TeX

`C_4^2._3C_4`
`% in TeX`

`G:=Group("C4^2.3C4");`
`// GroupNames label`

`G:=SmallGroup(64,37);`
`// by ID`

`G=gap.SmallGroup(64,37);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,332,158,681,255,117,1444]);`
`// Polycyclic`

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

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