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## G = 2+ 1+4.C6order 192 = 26·3

### 1st non-split extension by 2+ 1+4 of C6 acting faithfully

Aliases: 2+ 1+4.1C6, (C22×C4)⋊2A4, C23.7D4⋊C3, C23.2(C2×A4), C23⋊A4.1C2, C2.5(C24⋊C6), SmallGroup(192,202)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — 2+ 1+4.C6
 Chief series C1 — C2 — C23 — 2+ 1+4 — C23⋊A4 — 2+ 1+4.C6
 Lower central 2+ 1+4 — 2+ 1+4.C6
 Upper central C1 — C2

Generators and relations for 2+ 1+4.C6
G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=ece-1=a-1, ac=ca, ad=da, eae-1=acd, bc=cb, bd=db, ebe-1=a-1c, dcd=a2c, ede-1=a-1bd >

Character table of 2+ 1+4.C6

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 12A 12B 12C 12D size 1 1 6 12 16 16 4 4 12 24 16 16 16 16 16 16 ρ1 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 linear of order 2 ρ3 1 1 1 1 ζ3 ζ32 -1 -1 1 -1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 -1 -1 1 -1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 3 3 3 -1 0 0 -3 -3 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ8 3 3 3 -1 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ9 4 -4 0 0 1 1 2i -2i 0 0 -1 -1 i -i i -i complex faithful ρ10 4 -4 0 0 1 1 -2i 2i 0 0 -1 -1 -i i -i i complex faithful ρ11 4 -4 0 0 ζ32 ζ3 -2i 2i 0 0 ζ65 ζ6 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex faithful ρ12 4 -4 0 0 ζ3 ζ32 2i -2i 0 0 ζ6 ζ65 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex faithful ρ13 4 -4 0 0 ζ3 ζ32 -2i 2i 0 0 ζ6 ζ65 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex faithful ρ14 4 -4 0 0 ζ32 ζ3 2i -2i 0 0 ζ65 ζ6 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex faithful ρ15 6 6 -2 -2 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C6 ρ16 6 6 -2 2 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C6

Permutation representations of 2+ 1+4.C6
On 16 points - transitive group 16T426
Generators in S16
```(1 11 3 5)(2 14 4 8)(6 16 12 10)(7 15 13 9)
(1 5)(2 8)(3 11)(4 14)(6 10)(7 15)(9 13)(12 16)
(1 15 3 9)(2 6 4 12)(5 7 11 13)(8 10 14 16)
(1 3)(2 4)(5 11)(8 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,426);`

On 16 points - transitive group 16T428
Generators in S16
```(1 14 3 8)(2 9 4 15)(5 13 11 7)(6 16 12 10)
(2 13)(4 7)(5 9)(6 12)(8 14)(11 15)
(1 10 3 16)(2 9 4 15)(5 7 11 13)(6 8 12 14)
(1 16)(2 13)(3 10)(4 7)(5 15)(6 8)(9 11)(12 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,428);`

Matrix representation of 2+ 1+4.C6 in GL4(𝔽5) generated by

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

2+ 1+4.C6 in GAP, Magma, Sage, TeX

`2_+^{1+4}.C_6`
`% in TeX`

`G:=Group("ES+(2,2).C6");`
`// GroupNames label`

`G:=SmallGroup(192,202);`
`// by ID`

`G=gap.SmallGroup(192,202);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,1683,262,851,375,3540,1027]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=d^2=1,c^2=e^6=a^2,b*a*b=e*c*e^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*c,d*c*d=a^2*c,e*d*e^-1=a^-1*b*d>;`
`// generators/relations`

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