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## G = C23.7D6order 96 = 25·3

### 2nd non-split extension by C23 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23.7D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6.D4 — C23.7D6
 Lower central C3 — C6 — C2×C6 — C23.7D6
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C23.7D6
G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=ba=ab, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Character table of C23.7D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 2 2 2 4 2 4 12 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 -1 i i -i -i 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ8 1 1 -1 1 -1 1 1 -1 -i -i i i 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 -2 2 0 2 0 0 0 0 0 -2 2 -2 0 0 2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -1 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 2 -2 -1 -2 0 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 2 -2 -2 0 2 0 0 0 0 0 -2 2 -2 0 0 -2 2 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 -2 2 -1 -2 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 -2 2 -2 -2 -1 2 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 -2 -2 2 0 -1 0 0 0 0 0 1 -1 1 √-3 -√-3 -1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ16 2 2 2 -2 -2 0 -1 0 0 0 0 0 1 -1 1 -√-3 √-3 1 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ17 2 2 2 -2 -2 0 -1 0 0 0 0 0 1 -1 1 √-3 -√-3 1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 2 0 -1 0 0 0 0 0 1 -1 1 -√-3 √-3 -1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 0 4 0 0 0 0 0 0 -4 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√-3 2 2√-3 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 0 0 2√-3 2 -2√-3 0 0 0 0 0 0 complex faithful

Permutation representations of C23.7D6
On 24 points - transitive group 24T96
Generators in S24
```(2 13)(4 15)(6 17)(8 21)(10 23)(12 19)
(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)
(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 19)(2 11 13 24)(3 23)(4 9 15 22)(5 21)(6 7 17 20)(8 16)(10 14)(12 18)```

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

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

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

`G:=TransitiveGroup(24,96);`

On 24 points - transitive group 24T98
Generators in S24
```(1 20)(2 24)(3 22)(4 23)(5 21)(6 19)(7 14)(8 18)(9 16)(10 13)(11 17)(12 15)
(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 14 20 11)(2 16 24 10)(3 18 22 12)(4 17 23 7)(5 13 21 9)(6 15 19 8)```

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

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

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

`G:=TransitiveGroup(24,98);`

On 24 points - transitive group 24T111
Generators in S24
```(2 20)(4 22)(6 24)(7 16)(9 18)(11 14)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(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 6 12 14)(2 10 16 5)(3 4 8 18)(7 23 20 13)(9 21 22 17)(11 19 24 15)```

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

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

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

`G:=TransitiveGroup(24,111);`

Matrix representation of C23.7D6 in GL4(𝔽7) generated by

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

C23.7D6 in GAP, Magma, Sage, TeX

`C_2^3._7D_6`
`% in TeX`

`G:=Group("C2^3.7D6");`
`// GroupNames label`

`G:=SmallGroup(96,41);`
`// by ID`

`G=gap.SmallGroup(96,41);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,579,2309]);`
`// Polycyclic`

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

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