Copied to
clipboard

G = C23.12D6order 96 = 25·3

7th non-split extension by C23 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23.12D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — C23.12D6
 Lower central C3 — C2×C6 — C23.12D6
 Upper central C1 — C22 — C2×D4

Generators and relations for C23.12D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, dad-1=ab=ba, eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 162 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.4D4, C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, C23.12D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, D42S3 [×2], C2×C3⋊D4, C23.12D6

Character table of C23.12D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 4 4 2 2 2 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 0 0 2 2 -2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -1 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 -2 2 orthogonal lifted from D4 ρ12 2 2 2 2 -2 2 -1 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 -2 -2 -1 2 2 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ14 2 2 2 2 2 -2 -1 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 √-3 -√-3 √-3 -√-3 -1 1 complex lifted from C3⋊D4 ρ16 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 -√-3 √-3 -√-3 √-3 -1 1 complex lifted from C3⋊D4 ρ17 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 √-3 √-3 -√-3 -√-3 1 -1 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 -√-3 -√-3 √-3 √-3 1 -1 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 0 2 0 0 0 0 2i -2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 -2 2 0 0 2 0 0 0 0 -2i 2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 0 0 2 0 0 -2i 2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 0 0 2 0 0 2i -2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 -4 4 0 0 -2 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 0 -2 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 C23.12D6
On 48 points
Generators in S48
```(2 8)(4 10)(6 12)(13 26)(14 33)(15 28)(16 35)(17 30)(18 25)(19 32)(20 27)(21 34)(22 29)(23 36)(24 31)(37 43)(39 45)(41 47)
(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 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(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 35 40 22)(2 28 41 15)(3 33 42 20)(4 26 43 13)(5 31 44 18)(6 36 45 23)(7 29 46 16)(8 34 47 21)(9 27 48 14)(10 32 37 19)(11 25 38 24)(12 30 39 17)```

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

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

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

Matrix representation of C23.12D6 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 3 0 0 0 0 0 12 0 0 1 0
,
 0 3 0 0 4 0 0 0 0 0 5 0 0 0 0 5
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,0,1,0,0,12,0],[0,4,0,0,3,0,0,0,0,0,5,0,0,0,0,5] >;`

C23.12D6 in GAP, Magma, Sage, TeX

`C_2^3._{12}D_6`
`% in TeX`

`G:=Group("C2^3.12D6");`
`// GroupNames label`

`G:=SmallGroup(96,143);`
`// by ID`

`G=gap.SmallGroup(96,143);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,116,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,e*a*e^-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