Copied to
clipboard

## G = D24⋊C2order 96 = 25·3

### 5th semidirect product of D24 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D24⋊C2
 Chief series C1 — C3 — C6 — C12 — C4×S3 — Q8⋊3S3 — D24⋊C2
 Lower central C3 — C6 — C12 — D24⋊C2
 Upper central C1 — C2 — C4 — Q16

Generators and relations for D24⋊C2
G = < a,b,c | a24=b2=c2=1, bab=a-1, cac=a17, cbc=a4b >

Subgroups: 162 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3×Q8, C4○D8, S3×C8, D24, Q82S3, C3×Q16, Q83S3, D24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S3×D4, D24⋊C2

Character table of D24⋊C2

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 8A 8B 8C 8D 12A 12B 12C 24A 24B size 1 1 6 12 12 2 2 3 3 4 4 2 2 2 6 6 4 8 8 4 4 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 0 0 0 -1 2 0 0 -2 2 -1 -2 -2 0 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ10 2 2 0 0 0 -1 2 0 0 -2 -2 -1 2 2 0 0 -1 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 0 0 2 -2 -2 -2 0 0 2 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 2 -2 2 2 0 0 2 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 0 0 0 -1 2 0 0 2 2 -1 2 2 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 0 0 0 -1 2 0 0 2 -2 -1 -2 -2 0 0 -1 1 -1 1 1 orthogonal lifted from D6 ρ15 2 -2 0 0 0 2 0 2i -2i 0 0 -2 √2 -√2 -√-2 √-2 0 0 0 -√2 √2 complex lifted from C4○D8 ρ16 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 -√2 √2 -√-2 √-2 0 0 0 √2 -√2 complex lifted from C4○D8 ρ17 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 √2 -√2 √-2 -√-2 0 0 0 -√2 √2 complex lifted from C4○D8 ρ18 2 -2 0 0 0 2 0 2i -2i 0 0 -2 -√2 √2 √-2 -√-2 0 0 0 √2 -√2 complex lifted from C4○D8 ρ19 4 4 0 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 orthogonal lifted from S3×D4 ρ20 4 -4 0 0 0 -2 0 0 0 0 0 2 2√2 -2√2 0 0 0 0 0 √2 -√2 orthogonal faithful ρ21 4 -4 0 0 0 -2 0 0 0 0 0 2 -2√2 2√2 0 0 0 0 0 -√2 √2 orthogonal faithful

Smallest permutation representation of D24⋊C2
On 48 points
Generators in S48
```(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 48)(46 47)
(1 29)(2 46)(3 39)(4 32)(5 25)(6 42)(7 35)(8 28)(9 45)(10 38)(11 31)(12 48)(13 41)(14 34)(15 27)(16 44)(17 37)(18 30)(19 47)(20 40)(21 33)(22 26)(23 43)(24 36)```

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

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

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

Matrix representation of D24⋊C2 in GL4(𝔽7) generated by

 5 3 4 6 4 2 0 0 5 6 3 1 6 1 4 0
,
 6 6 1 1 6 4 4 4 5 3 0 1 1 0 5 4
,
 2 5 2 4 4 4 2 2 5 1 5 6 4 6 1 3
`G:=sub<GL(4,GF(7))| [5,4,5,6,3,2,6,1,4,0,3,4,6,0,1,0],[6,6,5,1,6,4,3,0,1,4,0,5,1,4,1,4],[2,4,5,4,5,4,1,6,2,2,5,1,4,2,6,3] >;`

D24⋊C2 in GAP, Magma, Sage, TeX

`D_{24}\rtimes C_2`
`% in TeX`

`G:=Group("D24:C2");`
`// GroupNames label`

`G:=SmallGroup(96,126);`
`// by ID`

`G=gap.SmallGroup(96,126);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,362,116,86,297,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^4*b>;`
`// generators/relations`

Export

׿
×
𝔽