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## G = C24.19D4order 192 = 26·3

### 19th non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.19D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C24.19D4
 Lower central C3 — C6 — C2×C12 — C24.19D4
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C24.19D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=a7, cac=a-1, cbc=a12b3 >

Subgroups: 416 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, S3×C8, C8⋊S3, C24⋊C2, D24, C4.Dic3, C2×C24, C3×M4(2), C2×D12, C4○D12, D4.4D4, C12.46D4, C3×C8.C4, C8○D12, C2×D24, C8⋊D6, C24.19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.4D4, C12⋊D4, C24.19D4

Character table of C24.19D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 12 24 24 2 2 2 12 2 4 2 2 4 8 8 12 12 2 2 4 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 -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 0 -1 2 2 0 -1 -1 -2 -2 -2 -2 2 0 0 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 0 0 0 -1 2 2 0 -1 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 0 2 -2 2 2 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 0 2 2 -2 0 2 -2 -2 -2 2 0 0 0 0 -2 -2 2 -2 2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 0 0 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 0 0 -1 2 2 0 -1 -1 2 2 2 -2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ15 2 2 2 0 0 0 -1 2 2 0 -1 -1 -2 -2 -2 2 -2 0 0 -1 -1 -1 1 1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ16 2 2 -2 0 0 0 2 2 -2 0 2 -2 2 2 -2 0 0 0 0 -2 -2 2 2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 0 0 0 -1 2 -2 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 1 -1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ18 2 2 -2 0 0 0 -1 2 -2 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 1 -1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ19 2 2 -2 0 0 0 -1 2 -2 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 -1 1 -1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ20 2 2 -2 0 0 0 -1 2 -2 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 -1 1 -1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ21 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 -2i 2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 2i -2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 0 -2 -4 4 0 -2 2 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 0 0 -2 -4 -4 0 -2 -2 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ25 4 -4 0 0 0 0 4 0 0 0 -4 0 -2√2 2√2 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ26 4 -4 0 0 0 0 4 0 0 0 -4 0 2√2 -2√2 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ27 4 -4 0 0 0 0 -2 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 2√3 -2√3 0 √2 √6 -√2 -√6 0 0 0 0 orthogonal faithful ρ28 4 -4 0 0 0 0 -2 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 2√3 -2√3 0 -√2 -√6 √2 √6 0 0 0 0 orthogonal faithful ρ29 4 -4 0 0 0 0 -2 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 -2√3 2√3 0 √2 -√6 -√2 √6 0 0 0 0 orthogonal faithful ρ30 4 -4 0 0 0 0 -2 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 -2√3 2√3 0 -√2 √6 √2 -√6 0 0 0 0 orthogonal faithful

Smallest permutation representation of C24.19D4
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 44 19 26 13 32 7 38)(2 27 20 33 14 39 8 45)(3 34 21 40 15 46 9 28)(4 41 22 47 16 29 10 35)(5 48 23 30 17 36 11 42)(6 31 24 37 18 43 12 25)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 24)(18 23)(19 22)(20 21)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)```

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

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

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

Matrix representation of C24.19D4 in GL6(𝔽73)

 9 45 0 0 0 0 0 65 0 0 0 0 0 0 13 25 0 0 0 0 70 28 0 0 0 0 0 18 28 50 0 0 4 53 35 13
,
 27 27 0 0 0 0 0 46 0 0 0 0 0 0 0 0 1 0 0 0 0 46 0 71 0 0 21 3 0 0 0 0 25 10 0 27
,
 9 45 0 0 0 0 55 64 0 0 0 0 0 0 13 25 0 0 0 0 40 60 0 0 0 0 0 18 28 50 0 0 44 53 15 45

`G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,45,65,0,0,0,0,0,0,13,70,0,4,0,0,25,28,18,53,0,0,0,0,28,35,0,0,0,0,50,13],[27,0,0,0,0,0,27,46,0,0,0,0,0,0,0,0,21,25,0,0,0,46,3,10,0,0,1,0,0,0,0,0,0,71,0,27],[9,55,0,0,0,0,45,64,0,0,0,0,0,0,13,40,0,44,0,0,25,60,18,53,0,0,0,0,28,15,0,0,0,0,50,45] >;`

C24.19D4 in GAP, Magma, Sage, TeX

`C_{24}._{19}D_4`
`% in TeX`

`G:=Group("C24.19D4");`
`// GroupNames label`

`G:=SmallGroup(192,456);`
`// by ID`

`G=gap.SmallGroup(192,456);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,1123,136,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=a^12*b^3>;`
`// generators/relations`

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