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## G = C23⋊2D6order 96 = 25·3

### 1st semidirect product of C23 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23⋊2D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6
 Lower central C3 — C2×C6 — C23⋊2D6
 Upper central C1 — C22 — C2×D4

Generators and relations for C232D6
G = < a,b,c,d,e | a2=b2=c2=d6=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 354 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×4], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], Dic3 [×2], C12, D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C22≀C2, D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C232D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, C232D6

Character table of C232D6

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 2 2 4 6 6 6 6 2 4 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 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 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 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 -1 -1 -1 linear of order 2 ρ5 1 1 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 ρ6 1 1 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 ρ7 1 1 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 ρ8 1 1 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 ρ9 2 2 2 2 2 2 -2 0 0 0 0 -1 -2 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 -2 0 0 2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 2 0 0 -2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 2 -2 0 0 0 0 2 -2 0 2 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 2 0 0 0 0 -1 -2 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ14 2 -2 2 -2 0 0 0 0 -2 2 0 2 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 -2 -2 -2 0 0 0 0 -1 2 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ16 2 2 -2 -2 2 -2 0 0 0 0 0 2 0 0 0 -2 2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 2 2 0 0 0 0 -1 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 2 -2 -2 -2 2 0 0 0 0 0 2 0 0 0 -2 2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 -2 2 0 0 0 0 0 -1 0 0 0 1 -1 1 -1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 2 -2 0 0 0 0 0 -1 0 0 0 1 -1 1 1 -1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 2 -2 0 0 0 0 0 -1 0 0 0 1 -1 1 1 -1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 -2 2 0 0 0 0 0 -1 0 0 0 1 -1 1 -1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 -2 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 -2 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4

Permutation representations of C232D6
On 24 points - transitive group 24T145
Generators in S24
```(1 15)(2 13)(3 17)(4 20)(5 24)(6 22)(7 16)(8 14)(9 18)(10 21)(11 19)(12 23)
(1 5)(2 6)(3 4)(7 11)(8 12)(9 10)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)(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 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)```

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

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

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

`G:=TransitiveGroup(24,145);`

Matrix representation of C232D6 in GL4(𝔽13) generated by

 2 4 0 0 9 11 0 0 0 0 0 3 0 0 9 0
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 1 1 0 0 12 0 0 0 0 0 12 0 0 0 0 1
,
 1 1 0 0 0 12 0 0 0 0 1 0 0 0 0 12
`G:=sub<GL(4,GF(13))| [2,9,0,0,4,11,0,0,0,0,0,9,0,0,3,0],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,0,12] >;`

C232D6 in GAP, Magma, Sage, TeX

`C_2^3\rtimes_2D_6`
`% in TeX`

`G:=Group("C2^3:2D6");`
`// GroupNames label`

`G:=SmallGroup(96,144);`
`// by ID`

`G=gap.SmallGroup(96,144);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,2309]);`
`// Polycyclic`

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

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