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## G = 2+ 1+4.3C6order 192 = 26·3

### The non-split extension by 2+ 1+4 of C6 acting via C6/C2=C3

Aliases: 2+ 1+4.3C6, C4○D42A4, (C22×C4)⋊3A4, Q8.2(C2×A4), C23⋊A44C2, C23.7(C2×A4), C4.2(C22⋊A4), C2.C252C3, C2.5(C2×C22⋊A4), SmallGroup(192,1509)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — 2+ 1+4.3C6
 Chief series C1 — C2 — C23 — 2+ 1+4 — C23⋊A4 — 2+ 1+4.3C6
 Lower central 2+ 1+4 — 2+ 1+4.3C6
 Upper central C1 — C4

Generators and relations for 2+ 1+4.3C6
G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=a-1, ac=ca, ad=da, eae-1=a2bc, bc=cb, bd=db, ebe-1=a2bcd, dcd=a2c, ece-1=a-1d, ede-1=a-1bd >

Subgroups: 567 in 165 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×5], C22 [×12], C6, C2×C4 [×20], D4 [×20], Q8 [×2], Q8 [×6], C23 [×3], C23 [×4], C12, A4 [×3], C22×C4 [×3], C22×C4 [×4], C2×D4 [×15], C2×Q8 [×5], C4○D4 [×2], C4○D4 [×26], SL2(𝔽3) [×2], C2×A4 [×3], C2×C4○D4 [×5], 2+ 1+4, 2+ 1+4 [×3], 2- 1+4 [×2], C4×A4 [×3], C4.A4 [×2], C2.C25, C23⋊A4, 2+ 1+4.3C6
Quotients: C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C22⋊A4, C2×C22⋊A4, 2+ 1+4.3C6

Character table of 2+ 1+4.3C6

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A 6B 12A 12B 12C 12D size 1 1 6 6 6 6 6 16 16 1 1 6 6 6 6 6 16 16 16 16 16 16 ρ1 1 1 1 1 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 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 1 1 1 -1 ζ3 ζ32 -1 -1 1 -1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ5 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 1 1 1 -1 ζ32 ζ3 -1 -1 1 -1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ7 3 3 -1 -1 -1 -1 3 0 0 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ8 3 3 3 -1 -1 -1 -1 0 0 3 3 -1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ9 3 3 1 -1 -1 -1 -3 0 0 -3 -3 3 1 1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ10 3 3 -3 -1 -1 -1 1 0 0 -3 -3 -1 1 1 1 3 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ11 3 3 -1 3 -1 -1 -1 0 0 3 3 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ12 3 3 1 -1 3 -1 1 0 0 -3 -3 -1 1 1 -3 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ13 3 3 1 -1 -1 3 1 0 0 -3 -3 -1 -3 1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 1 3 -1 -1 1 0 0 -3 -3 -1 1 -3 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 -1 -1 3 -1 -1 0 0 3 3 -1 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ16 3 3 -1 -1 -1 3 -1 0 0 3 3 -1 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ17 4 -4 0 0 0 0 0 1 1 -4i 4i 0 0 0 0 0 -1 -1 i i -i -i complex faithful ρ18 4 -4 0 0 0 0 0 1 1 4i -4i 0 0 0 0 0 -1 -1 -i -i i i complex faithful ρ19 4 -4 0 0 0 0 0 ζ32 ζ3 -4i 4i 0 0 0 0 0 ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 complex faithful ρ20 4 -4 0 0 0 0 0 ζ3 ζ32 -4i 4i 0 0 0 0 0 ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 complex faithful ρ21 4 -4 0 0 0 0 0 ζ3 ζ32 4i -4i 0 0 0 0 0 ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 complex faithful ρ22 4 -4 0 0 0 0 0 ζ32 ζ3 4i -4i 0 0 0 0 0 ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 complex faithful

Permutation representations of 2+ 1+4.3C6
On 16 points - transitive group 16T423
Generators in S16
```(1 7 3 13)(2 16 4 10)(5 15 11 9)(6 14 12 8)
(1 13)(2 10)(3 7)(4 16)(5 9)(6 14)(8 12)(11 15)
(1 5 3 11)(2 14 4 8)(6 16 12 10)(7 15 13 9)
(5 11)(6 12)(8 14)(9 15)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,423);`

Matrix representation of 2+ 1+4.3C6 in GL4(𝔽5) generated by

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

2+ 1+4.3C6 in GAP, Magma, Sage, TeX

`2_+^{1+4}._3C_6`
`% in TeX`

`G:=Group("ES+(2,2).3C6");`
`// GroupNames label`

`G:=SmallGroup(192,1509);`
`// by ID`

`G=gap.SmallGroup(192,1509);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,375,1524,1027]);`
`// Polycyclic`

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

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