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## G = C23.6D6order 96 = 25·3

### 1st non-split extension by C23 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23.6D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — C23.6D6
 Lower central C3 — C6 — C2×C6 — C23.6D6
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for C23.6D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd5 >

Character table of C23.6D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 2 2 2 12 2 4 4 12 12 12 2 2 2 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 ρ4 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 ρ5 1 1 -1 1 -1 -1 1 -i i i -i 1 1 -1 -1 1 -1 -i -i i i linear of order 4 ρ6 1 1 -1 1 -1 1 1 -i i -i i -1 1 -1 -1 1 -1 -i -i i i linear of order 4 ρ7 1 1 -1 1 -1 -1 1 i -i -i i 1 1 -1 -1 1 -1 i i -i -i linear of order 4 ρ8 1 1 -1 1 -1 1 1 i -i i -i -1 1 -1 -1 1 -1 i i -i -i linear of order 4 ρ9 2 2 2 -2 -2 0 2 0 0 0 0 0 2 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 2 0 0 0 0 0 2 -2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 2 2 0 -1 2 2 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 -2 2 0 -1 0 0 0 0 0 -1 1 1 1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ14 2 2 -2 -2 2 0 -1 0 0 0 0 0 -1 1 1 1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ15 2 2 -2 2 -2 0 -1 -2i 2i 0 0 0 -1 1 1 -1 1 i i -i -i complex lifted from C4×S3 ρ16 2 2 -2 2 -2 0 -1 2i -2i 0 0 0 -1 1 1 -1 1 -i -i i i complex lifted from C4×S3 ρ17 2 2 2 -2 -2 0 -1 0 0 0 0 0 -1 -1 -1 1 1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 -2 -2 0 -1 0 0 0 0 0 -1 -1 -1 1 1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 -4 0 0 0 0 -2 0 0 0 0 0 2 -2√-3 2√-3 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 0 0 2 2√-3 -2√-3 0 0 0 0 0 0 complex faithful

Permutation representations of C23.6D6
On 24 points - transitive group 24T94
Generators in S24
```(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 9)(3 11)(5 7)(13 19)(15 21)(17 23)
(1 9)(2 10)(3 11)(4 12)(5 7)(6 8)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 22)(2 15 10 21)(3 14)(4 19 12 13)(5 18)(6 23 8 17)(7 24)(9 16)(11 20)```

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

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

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

`G:=TransitiveGroup(24,94);`

On 24 points - transitive group 24T103
Generators in S24
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 22)(4 24)(6 14)(8 16)(10 18)(12 20)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(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)
(1 6 15 12)(2 19 8 5)(3 24 17 18)(4 9 10 23)(7 20 21 14)(11 16 13 22)```

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

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

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

`G:=TransitiveGroup(24,103);`

On 24 points - transitive group 24T108
Generators in S24
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 22)(2 8)(3 24)(4 10)(5 14)(6 12)(7 16)(9 18)(11 20)(13 19)(15 21)(17 23)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 9)(2 23 17 8)(3 22)(4 6 19 21)(7 18)(10 15 13 12)(11 14)(16 24)```

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

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

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

`G:=TransitiveGroup(24,108);`

On 24 points - transitive group 24T119
Generators in S24
```(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 10)(2 8)(3 12)(4 7)(5 11)(6 9)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 17 7 20)(2 13 11 16)(3 21 9 24)(4 23 10 14)(5 19 8 22)(6 15 12 18)```

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

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

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

`G:=TransitiveGroup(24,119);`

Matrix representation of C23.6D6 in GL4(𝔽7) generated by

 0 1 4 5 1 0 3 5 0 0 1 0 0 0 0 6
,
 1 5 2 6 1 5 5 3 0 0 1 0 5 2 1 0
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 5 2 2 1 6 1 5 5 5 4 4 3 5 0
,
 4 2 2 6 3 0 5 4 6 1 4 3 4 4 5 6
`G:=sub<GL(4,GF(7))| [0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[1,1,0,5,5,5,0,2,2,5,1,1,6,3,0,0],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,2,5,4,0,1,5,3,5,6,5,5,2,1,4,0],[4,3,6,4,2,0,1,4,2,5,4,5,6,4,3,6] >;`

C23.6D6 in GAP, Magma, Sage, TeX

`C_2^3._6D_6`
`% in TeX`

`G:=Group("C2^3.6D6");`
`// GroupNames label`

`G:=SmallGroup(96,13);`
`// by ID`

`G=gap.SmallGroup(96,13);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,297,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;`
`// generators/relations`

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