Copied to
clipboard

## G = D24.C4order 192 = 26·3

### 4th non-split extension by D24 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D24.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×D24 — D24.C4
 Lower central C3 — C6 — C12 — C24 — D24.C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for D24.C4
G = < a,b,c | a24=b2=1, c4=a12, bab=a-1, cac-1=a19, cbc-1=a15b >

Subgroups: 288 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, D4, C23, C12, D6, C2×C6, C16, C2×C8, M4(2), D8, C2×D4, C24, C24, D12, C2×C12, C22×S3, C8.C4, M5(2), C2×D8, C3⋊C16, D24, D24, C2×C24, C3×M4(2), C2×D12, M5(2)⋊C2, C12.C8, C3×C8.C4, C2×D24, D24.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D6⋊C4, D4⋊S3, Q82S3, M5(2)⋊C2, C6.D8, D24.C4

Character table of D24.C4

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 8A 8B 8C 8D 8E 12A 12B 12C 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 24 24 2 2 2 2 4 2 2 4 8 8 2 2 4 12 12 12 12 4 4 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 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 i -i -1 -1 1 -i i -i i -1 1 -1 1 -i i i -i linear of order 4 ρ6 1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -i i -1 -1 1 -i i -i i -1 1 -1 1 i -i -i i linear of order 4 ρ7 1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 i -i -1 -1 1 i -i i -i -1 1 -1 1 -i i i -i linear of order 4 ρ8 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -i i -1 -1 1 i -i i -i -1 1 -1 1 i -i -i i linear of order 4 ρ9 2 2 2 0 0 2 2 2 2 2 -2 -2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 2 -2 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 -1 2 2 -1 -1 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 0 0 -1 2 2 -1 -1 2 2 2 -2 -2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 0 0 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 √2 √2 -√2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 -2 0 0 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 -√2 -√2 √2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ15 2 2 -2 0 0 -1 -2 2 -1 1 2 2 -2 0 0 1 1 -1 0 0 0 0 -1 1 -1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ16 2 2 -2 0 0 -1 -2 2 -1 1 2 2 -2 0 0 1 1 -1 0 0 0 0 -1 1 -1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ17 2 2 -2 0 0 -1 -2 2 -1 1 -2 -2 2 2i -2i 1 1 -1 0 0 0 0 1 -1 1 -1 i -i -i i complex lifted from C4×S3 ρ18 2 2 -2 0 0 -1 -2 2 -1 1 -2 -2 2 -2i 2i 1 1 -1 0 0 0 0 1 -1 1 -1 -i i i -i complex lifted from C4×S3 ρ19 2 2 2 0 0 2 -2 -2 2 2 0 0 0 0 0 -2 -2 -2 -√-2 √-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 2 2 0 0 2 -2 -2 2 2 0 0 0 0 0 -2 -2 -2 √-2 -√-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 2 2 0 0 -1 2 2 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 2 0 0 -1 2 2 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 4 4 0 0 -2 -4 -4 -2 -2 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ24 4 4 -4 0 0 -2 4 -4 -2 2 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 -4 0 0 0 4 0 0 -4 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ26 4 -4 0 0 0 4 0 0 -4 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ27 4 -4 0 0 0 -2 0 0 2 0 2√2 -2√2 0 0 0 -2√3 2√3 0 0 0 0 0 -√2 √6 √2 -√6 0 0 0 0 orthogonal faithful ρ28 4 -4 0 0 0 -2 0 0 2 0 -2√2 2√2 0 0 0 2√3 -2√3 0 0 0 0 0 √2 √6 -√2 -√6 0 0 0 0 orthogonal faithful ρ29 4 -4 0 0 0 -2 0 0 2 0 2√2 -2√2 0 0 0 2√3 -2√3 0 0 0 0 0 -√2 -√6 √2 √6 0 0 0 0 orthogonal faithful ρ30 4 -4 0 0 0 -2 0 0 2 0 -2√2 2√2 0 0 0 -2√3 2√3 0 0 0 0 0 √2 -√6 -√2 √6 0 0 0 0 orthogonal faithful

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

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

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

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

Matrix representation of D24.C4 in GL4(𝔽97) generated by

 18 16 0 0 81 2 0 0 12 0 16 2 12 12 95 18
,
 79 95 0 0 16 18 0 0 60 25 68 58 25 62 29 29
,
 56 15 33 33 82 41 31 33 53 54 56 82 96 53 15 41
`G:=sub<GL(4,GF(97))| [18,81,12,12,16,2,0,12,0,0,16,95,0,0,2,18],[79,16,60,25,95,18,25,62,0,0,68,29,0,0,58,29],[56,82,53,96,15,41,54,53,33,31,56,15,33,33,82,41] >;`

D24.C4 in GAP, Magma, Sage, TeX

`D_{24}.C_4`
`% in TeX`

`G:=Group("D24.C4");`
`// GroupNames label`

`G:=SmallGroup(192,54);`
`// by ID`

`G=gap.SmallGroup(192,54);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,184,675,794,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^2=1,c^4=a^12,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^15*b>;`
`// generators/relations`

Export

׿
×
𝔽