Copied to
clipboard

## G = C4○D24order 96 = 25·3

### Central product of C4 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4○D24
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — C4○D24
 Lower central C3 — C6 — C12 — C4○D24
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C4○D24
G = < a,b,c | a4=c2=1, b12=a2, ab=ba, ac=ca, cbc=a2b11 >

Subgroups: 162 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4○D8, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C4○D24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, C2×D12, C4○D24

Character table of C4○D24

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 12 12 2 1 1 2 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 -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 -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 -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 -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 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 1 1 -1 -1 1 -1 linear of order 2 ρ9 2 2 2 0 0 2 -2 -2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 -1 2 2 2 0 0 -1 -1 -1 -2 -2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 0 0 -1 -2 -2 2 0 0 -1 1 1 2 -2 -2 2 -1 -1 1 1 1 -1 1 1 -1 -1 1 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 -1 -2 -2 2 0 0 -1 1 1 -2 2 2 -2 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 orthogonal lifted from D6 ρ13 2 2 -2 0 0 2 2 2 -2 0 0 2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 0 -1 2 2 2 0 0 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 1 0 0 0 0 1 1 -1 -1 -√3 √3 √3 -√3 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ16 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 -√3 -√3 √3 -√3 √3 -√3 √3 √3 orthogonal lifted from D12 ρ17 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 √3 √3 -√3 √3 -√3 √3 -√3 -√3 orthogonal lifted from D12 ρ18 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 1 0 0 0 0 1 1 -1 -1 √3 -√3 -√3 √3 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ19 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 √-2 -√2 √2 -√-2 0 0 2i -2i √2 -√-2 -√2 -√2 -√-2 √-2 √2 √-2 complex lifted from C4○D8 ρ20 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 √-2 √2 -√2 -√-2 0 0 -2i 2i -√2 -√-2 √2 √2 -√-2 √-2 -√2 √-2 complex lifted from C4○D8 ρ21 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 -√-2 -√2 √2 √-2 0 0 -2i 2i √2 √-2 -√2 -√2 √-2 -√-2 √2 -√-2 complex lifted from C4○D8 ρ22 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 -√-2 √2 -√2 √-2 0 0 2i -2i -√2 √-2 √2 √2 √-2 -√-2 -√2 -√-2 complex lifted from C4○D8 ρ23 2 -2 0 0 0 -1 -2i 2i 0 0 0 1 -√-3 √-3 -√-2 √2 -√2 √-2 √3 -√3 -i i ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 complex faithful ρ24 2 -2 0 0 0 -1 -2i 2i 0 0 0 1 -√-3 √-3 √-2 -√2 √2 -√-2 √3 -√3 -i i ζ83ζ32+ζ83+ζ8ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ87ζ3+ζ87-ζ85ζ3 complex faithful ρ25 2 -2 0 0 0 -1 -2i 2i 0 0 0 1 √-3 -√-3 √-2 -√2 √2 -√-2 -√3 √3 -i i ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 complex faithful ρ26 2 -2 0 0 0 -1 2i -2i 0 0 0 1 -√-3 √-3 √-2 √2 -√2 -√-2 -√3 √3 i -i ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ87-ζ85ζ32 complex faithful ρ27 2 -2 0 0 0 -1 2i -2i 0 0 0 1 √-3 -√-3 √-2 √2 -√2 -√-2 √3 -√3 i -i ζ83ζ3+ζ8ζ3+ζ8 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ87ζ3+ζ87-ζ85ζ3 complex faithful ρ28 2 -2 0 0 0 -1 2i -2i 0 0 0 1 √-3 -√-3 -√-2 -√2 √2 √-2 √3 -√3 i -i ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ83-ζ8ζ3 complex faithful ρ29 2 -2 0 0 0 -1 2i -2i 0 0 0 1 -√-3 √-3 -√-2 -√2 √2 √-2 -√3 √3 i -i ζ87ζ32+ζ85ζ32+ζ85 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ83ζ32+ζ83-ζ8ζ32 complex faithful ρ30 2 -2 0 0 0 -1 -2i 2i 0 0 0 1 √-3 -√-3 -√-2 √2 -√2 √-2 -√3 √3 -i i ζ87ζ3+ζ87+ζ85ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ83ζ32+ζ83-ζ8ζ32 complex faithful

Smallest permutation representation of C4○D24
On 48 points
Generators in S48
```(1 45 13 33)(2 46 14 34)(3 47 15 35)(4 48 16 36)(5 25 17 37)(6 26 18 38)(7 27 19 39)(8 28 20 40)(9 29 21 41)(10 30 22 42)(11 31 23 43)(12 32 24 44)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)```

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

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

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

Matrix representation of C4○D24 in GL2(𝔽73) generated by

 27 0 0 27
,
 55 50 23 5
,
 66 7 14 7
`G:=sub<GL(2,GF(73))| [27,0,0,27],[55,23,50,5],[66,14,7,7] >;`

C4○D24 in GAP, Magma, Sage, TeX

`C_4\circ D_{24}`
`% in TeX`

`G:=Group("C4oD24");`
`// GroupNames label`

`G:=SmallGroup(96,111);`
`// by ID`

`G=gap.SmallGroup(96,111);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,50,579,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=c^2=1,b^12=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^11>;`
`// generators/relations`

Export

׿
×
𝔽