Copied to
clipboard

## G = C23.8D6order 96 = 25·3

### 3rd non-split extension by C23 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23.8D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — C23.8D6
 Lower central C3 — C2×C6 — C23.8D6
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C23.8D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 122 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C422C2, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C23.8D6
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4 [×3], C22×S3, C422C2, C4○D12, D42S3 [×2], C23.8D6

Character table of C23.8D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 1 1 4 2 2 2 4 6 6 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 -1 2 2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 -1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 2 -1 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 -2 -1 -2 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ12 2 2 2 2 2 -1 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 0 2 -2i 2i 0 0 0 0 0 0 0 -2 -2 2 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ14 2 -2 -2 2 0 2 0 0 0 -2i 2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 2 -2 0 2 2i -2i 0 0 0 0 0 0 0 -2 -2 2 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ16 2 2 -2 -2 0 2 0 0 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 2 0 0 0 2i -2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 0 2 0 0 0 0 0 2i -2i 0 0 -2 2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 2 -2 0 -1 2i -2i 0 0 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 -1 2i -2i 0 0 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 -1 -2i 2i 0 0 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 -1 -2i 2i 0 0 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 -2 0 0 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ24 4 -4 -4 4 0 -2 0 0 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 C23.8D6
On 48 points
Generators in S48
```(2 42)(4 44)(6 46)(8 48)(10 38)(12 40)(13 27)(14 20)(15 29)(16 22)(17 31)(18 24)(19 33)(21 35)(23 25)(26 32)(28 34)(30 36)
(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 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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 26 47 24)(2 31 48 17)(3 36 37 22)(4 29 38 15)(5 34 39 20)(6 27 40 13)(7 32 41 18)(8 25 42 23)(9 30 43 16)(10 35 44 21)(11 28 45 14)(12 33 46 19)```

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

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

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

Matrix representation of C23.8D6 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 12 12
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 6 0 0 0 0 2 0 0 0 0 5 10 0 0 0 8
,
 0 2 0 0 6 0 0 0 0 0 12 11 0 0 0 1
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,2,0,0,0,0,5,0,0,0,10,8],[0,6,0,0,2,0,0,0,0,0,12,0,0,0,11,1] >;`

C23.8D6 in GAP, Magma, Sage, TeX

`C_2^3._8D_6`
`% in TeX`

`G:=Group("C2^3.8D6");`
`// GroupNames label`

`G:=SmallGroup(96,86);`
`// by ID`

`G=gap.SmallGroup(96,86);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,188,2309]);`
`// Polycyclic`

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

Export

׿
×
𝔽