Copied to
clipboard

G = C23.3D12order 192 = 26·3

3rd non-split extension by C23 of D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C23.3D12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×D4 — C23⋊2D6 — C23.3D12
 Lower central C3 — C6 — C2×C6 — C22×C6 — C23.3D12
 Upper central C1 — C2 — C22 — C2×D4 — C4.D4

Generators and relations for C23.3D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd11 >

Subgroups: 432 in 94 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], S3 [×2], C6, C6 [×3], C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], Dic3 [×2], C12, D6 [×8], C2×C6, C2×C6 [×4], C22⋊C4 [×3], M4(2), C2×D4, C2×D4, C24, C24, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C3×D4, C22×S3 [×4], C22×C6 [×2], C23⋊C4, C4.D4, C22≀C2, D6⋊C4, C6.D4, C6.D4, C3×M4(2), C2×C3⋊D4, C6×D4, S3×C23, C2≀C4, C23.7D6, C3×C4.D4, C232D6, C23.3D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C2≀C4, C23.6D6, C23.3D12

Character table of C23.3D12

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 12A 12B 24A 24B 24C 24D size 1 1 2 4 4 12 12 2 4 24 24 24 2 4 8 8 8 8 4 4 8 8 8 8 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 -i i -1 1 1 -1 1 -i i -1 -1 i -i -i i linear of order 4 ρ6 1 1 1 1 -1 1 1 1 -1 i -i -1 1 1 -1 1 i -i -1 -1 -i i i -i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 -1 -i i 1 1 1 -1 1 i -i -1 -1 -i i i -i linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 -1 i -i 1 1 1 -1 1 -i i -1 -1 i -i -i i linear of order 4 ρ9 2 2 2 -2 2 0 0 2 -2 0 0 0 2 2 2 -2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 0 0 -1 2 0 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -2 -2 0 0 2 2 0 0 0 2 2 -2 -2 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 0 0 -1 2 0 0 0 -1 -1 -1 -1 -2 -2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 2 -2 2 0 0 -1 -2 0 0 0 -1 -1 -1 1 0 0 1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ14 2 2 2 -2 2 0 0 -1 -2 0 0 0 -1 -1 -1 1 0 0 1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ15 2 2 2 2 -2 0 0 -1 -2 0 0 0 -1 -1 1 -1 -2i 2i 1 1 -i i i -i complex lifted from C4×S3 ρ16 2 2 2 2 -2 0 0 -1 -2 0 0 0 -1 -1 1 -1 2i -2i 1 1 i -i -i i complex lifted from C4×S3 ρ17 2 2 2 -2 -2 0 0 -1 2 0 0 0 -1 -1 1 1 0 0 -1 -1 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 -2 -2 0 0 -1 2 0 0 0 -1 -1 1 1 0 0 -1 -1 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 2 -2 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ20 4 -4 0 0 0 -2 2 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ21 4 4 -4 0 0 0 0 4 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 0 -2 0 0 0 0 -2 2 0 0 0 0 2√-3 -2√-3 0 0 0 0 complex lifted from C23.6D6 ρ23 4 4 -4 0 0 0 0 -2 0 0 0 0 -2 2 0 0 0 0 -2√-3 2√-3 0 0 0 0 complex lifted from C23.6D6 ρ24 8 -8 0 0 0 0 0 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23.3D12
On 24 points - transitive group 24T338
Generators in S24
```(1 7)(2 20)(3 21)(4 10)(5 11)(6 24)(8 14)(9 15)(12 18)(13 19)(16 22)(17 23)
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 6 7 24)(2 11 20 5)(3 4 21 10)(8 17 14 23)(9 22 15 16)(12 13 18 19)```

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

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

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

`G:=TransitiveGroup(24,338);`

On 24 points - transitive group 24T342
Generators in S24
```(2 14)(3 15)(6 18)(7 19)(10 22)(11 23)
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 24)(2 11 14 23)(3 22 15 10)(4 21)(5 20)(6 7 18 19)(8 17)(9 16)(12 13)```

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

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

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

`G:=TransitiveGroup(24,342);`

Matrix representation of C23.3D12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 72 0 72 72
,
 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 72 0 0 0 72 72 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 30 43 0 0 0 0 30 60 0 0 0 0 0 0 0 0 1 0 0 0 72 72 72 71 0 0 0 72 0 0 0 0 0 1 0 1
,
 30 43 0 0 0 0 13 43 0 0 0 0 0 0 0 0 72 0 0 0 1 1 1 2 0 0 72 0 0 0 0 0 0 72 0 72

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,72,0,0,0,1,0,72,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,0,72,0,0,0,0,0,72,72,1,0,0,1,72,0,0,0,0,0,71,0,1],[30,13,0,0,0,0,43,43,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0,72,0,0,72,1,0,0,0,0,0,2,0,72] >;`

C23.3D12 in GAP, Magma, Sage, TeX

`C_2^3._3D_{12}`
`% in TeX`

`G:=Group("C2^3.3D12");`
`// GroupNames label`

`G:=SmallGroup(192,34);`
`// by ID`

`G=gap.SmallGroup(192,34);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,346,297,851,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^11>;`
`// generators/relations`

Export

׿
×
𝔽