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## G = C62.C4order 144 = 24·32

### 2nd non-split extension by C62 of C4 acting faithfully

Aliases: C62.2C4, C324M4(2), C322C84C2, C3⋊Dic3.6C4, C22.(C32⋊C4), C3⋊Dic3.10C22, (C3×C6).6(C2×C4), C2.6(C2×C32⋊C4), (C2×C3⋊Dic3).7C2, SmallGroup(144,135)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C62.C4
 Lower central C32 — C3×C6 — C62.C4
 Upper central C1 — C2 — C22

Generators and relations for C62.C4
G = < a,b,c | a6=b6=1, c4=b3, ab=ba, cac-1=a-1b, cbc-1=a4b >

Character table of C62.C4

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D size 1 1 2 4 4 9 9 18 4 4 4 4 4 4 18 18 18 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 1 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 1 1 1 1 i -i i -i linear of order 4 ρ7 1 1 -1 1 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 1 1 1 1 -i i -i i linear of order 4 ρ9 2 -2 0 2 2 2i -2i 0 0 -2 0 0 0 -2 0 0 0 0 complex lifted from M4(2) ρ10 2 -2 0 2 2 -2i 2i 0 0 -2 0 0 0 -2 0 0 0 0 complex lifted from M4(2) ρ11 4 4 -4 -2 1 0 0 0 2 1 -1 -1 2 -2 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ12 4 4 4 -2 1 0 0 0 -2 1 1 1 -2 -2 0 0 0 0 orthogonal lifted from C32⋊C4 ρ13 4 4 4 1 -2 0 0 0 1 -2 -2 -2 1 1 0 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 4 -4 1 -2 0 0 0 -1 -2 2 2 -1 1 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ15 4 -4 0 1 -2 0 0 0 -3 2 0 0 3 -1 0 0 0 0 symplectic faithful, Schur index 2 ρ16 4 -4 0 -2 1 0 0 0 0 -1 -3 3 0 2 0 0 0 0 symplectic faithful, Schur index 2 ρ17 4 -4 0 1 -2 0 0 0 3 2 0 0 -3 -1 0 0 0 0 symplectic faithful, Schur index 2 ρ18 4 -4 0 -2 1 0 0 0 0 -1 3 -3 0 2 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C62.C4
On 24 points - transitive group 24T207
Generators in S24
```(1 9 21)(2 6)(3 23 11)(4 8)(5 13 17)(7 19 15)(10 14)(12 16)(18 22)(20 24)
(1 13 21 5 9 17)(2 18 10 6 22 14)(3 19 11 7 23 15)(4 16 24 8 12 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)```

`G:=sub<Sym(24)| (1,9,21)(2,6)(3,23,11)(4,8)(5,13,17)(7,19,15)(10,14)(12,16)(18,22)(20,24), (1,13,21,5,9,17)(2,18,10,6,22,14)(3,19,11,7,23,15)(4,16,24,8,12,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;`

`G:=Group( (1,9,21)(2,6)(3,23,11)(4,8)(5,13,17)(7,19,15)(10,14)(12,16)(18,22)(20,24), (1,13,21,5,9,17)(2,18,10,6,22,14)(3,19,11,7,23,15)(4,16,24,8,12,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );`

`G=PermutationGroup([[(1,9,21),(2,6),(3,23,11),(4,8),(5,13,17),(7,19,15),(10,14),(12,16),(18,22),(20,24)], [(1,13,21,5,9,17),(2,18,10,6,22,14),(3,19,11,7,23,15),(4,16,24,8,12,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,207);`

C62.C4 is a maximal subgroup of
Dic3≀C2  C62.2Q8  C3⋊Dic3.D4  (C2×C62).C4  C62.12D4  C62.15D4  C3⋊S3⋊M4(2)  C62.(C2×C4)  C33⋊M4(2)  C3312M4(2)
C62.C4 is a maximal quotient of
C322C8⋊C4  C325(C4⋊C8)  C623C8  He34M4(2)  C33⋊M4(2)  C3312M4(2)

Matrix representation of C62.C4 in GL4(𝔽5) generated by

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

C62.C4 in GAP, Magma, Sage, TeX

`C_6^2.C_4`
`% in TeX`

`G:=Group("C6^2.C4");`
`// GroupNames label`

`G:=SmallGroup(144,135);`
`// by ID`

`G=gap.SmallGroup(144,135);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,50,3364,256,4613,881]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^6=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;`
`// generators/relations`

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