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## G = C23.9D6order 96 = 25·3

### 4th non-split extension by C23 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23.9D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C23.9D6
 Lower central C3 — C2×C6 — C23.9D6
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C23.9D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=e2=b, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 186 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C23.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D12, S3×D4, D42S3, C23.9D6

Character table of C23.9D6

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 1 1 4 6 6 2 2 2 4 6 6 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 0 0 -1 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 -2 2 -2 0 -2 2 2 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 0 0 -1 2 2 -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 0 -1 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 2 2 0 0 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ14 2 -2 2 -2 0 2 -2 2 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 0 0 0 2 0 0 0 -2i 2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ16 2 -2 -2 2 0 0 0 2 0 0 0 2i -2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 2 -2 -2 0 0 0 2 2i -2i 0 0 0 0 0 -2 -2 2 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ18 2 2 -2 -2 0 0 0 2 -2i 2i 0 0 0 0 0 -2 -2 2 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ19 2 2 -2 -2 0 0 0 -1 2i -2i 0 0 0 0 0 1 1 -1 -√-3 √-3 -√3 -i √3 i complex lifted from C4○D12 ρ20 2 2 -2 -2 0 0 0 -1 2i -2i 0 0 0 0 0 1 1 -1 √-3 -√-3 √3 -i -√3 i complex lifted from C4○D12 ρ21 2 2 -2 -2 0 0 0 -1 -2i 2i 0 0 0 0 0 1 1 -1 -√-3 √-3 √3 i -√3 -i complex lifted from C4○D12 ρ22 2 2 -2 -2 0 0 0 -1 -2i 2i 0 0 0 0 0 1 1 -1 √-3 -√-3 -√3 i √3 -i complex lifted from C4○D12 ρ23 4 -4 4 -4 0 0 0 -2 0 0 0 0 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 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of C23.9D6
On 48 points
Generators in S48
```(1 44)(2 23)(3 46)(4 13)(5 48)(6 15)(7 38)(8 17)(9 40)(10 19)(11 42)(12 21)(14 34)(16 36)(18 26)(20 28)(22 30)(24 32)(25 39)(27 41)(29 43)(31 45)(33 47)(35 37)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 34 31 28)(26 27 32 33)(29 30 35 36)(37 44 43 38)(39 42 45 48)(40 47 46 41)```

`G:=sub<Sym(48)| (1,44)(2,23)(3,46)(4,13)(5,48)(6,15)(7,38)(8,17)(9,40)(10,19)(11,42)(12,21)(14,34)(16,36)(18,26)(20,28)(22,30)(24,32)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41)>;`

`G:=Group( (1,44)(2,23)(3,46)(4,13)(5,48)(6,15)(7,38)(8,17)(9,40)(10,19)(11,42)(12,21)(14,34)(16,36)(18,26)(20,28)(22,30)(24,32)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41) );`

`G=PermutationGroup([[(1,44),(2,23),(3,46),(4,13),(5,48),(6,15),(7,38),(8,17),(9,40),(10,19),(11,42),(12,21),(14,34),(16,36),(18,26),(20,28),(22,30),(24,32),(25,39),(27,41),(29,43),(31,45),(33,47),(35,37)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,47),(14,48),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,34,31,28),(26,27,32,33),(29,30,35,36),(37,44,43,38),(39,42,45,48),(40,47,46,41)]])`

Matrix representation of C23.9D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 1 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 12

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;`

C23.9D6 in GAP, Magma, Sage, TeX

`C_2^3._9D_6`
`% in TeX`

`G:=Group("C2^3.9D6");`
`// GroupNames label`

`G:=SmallGroup(96,90);`
`// by ID`

`G=gap.SmallGroup(96,90);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,2309]);`
`// Polycyclic`

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

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