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## G = D24⋊8C4order 192 = 26·3

### 8th semidirect product of D24 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D24⋊8C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C4○D24 — D24⋊8C4
 Lower central C3 — C6 — C12 — C24 — D24⋊8C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4.Q8

Generators and relations for D248C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a15b >

Subgroups: 208 in 58 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C4.Q8, M5(2), C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C3×C4⋊C4, C2×C24, C4○D12, D82C4, C12.C8, C3×C4.Q8, C4○D24, D248C4
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, D82C4, C6.D8, D248C4

Character table of D248C4

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

Smallest permutation representation of D248C4
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)
(2 20)(3 15)(4 10)(6 24)(7 19)(8 14)(11 23)(12 18)(16 22)(25 28 37 40)(26 47 38 35)(27 42 39 30)(29 32 41 44)(31 46 43 34)(33 36 45 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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43), (2,20)(3,15)(4,10)(6,24)(7,19)(8,14)(11,23)(12,18)(16,22)(25,28,37,40)(26,47,38,35)(27,42,39,30)(29,32,41,44)(31,46,43,34)(33,36,45,48)>;`

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

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

Matrix representation of D248C4 in GL6(𝔽97)

 0 1 0 0 0 0 96 96 0 0 0 0 0 0 40 40 0 0 0 0 57 40 0 0 0 0 0 0 57 40 0 0 0 0 57 57
,
 27 42 0 0 0 0 15 70 0 0 0 0 0 0 0 0 57 40 0 0 0 0 57 57 0 0 40 40 0 0 0 0 57 40 0 0
,
 75 0 0 0 0 0 0 75 0 0 0 0 0 0 1 0 0 0 0 0 0 96 0 0 0 0 0 0 40 57 0 0 0 0 57 57

`G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,0,0,0,0,57,57,0,0,0,0,40,57],[27,15,0,0,0,0,42,70,0,0,0,0,0,0,0,0,40,57,0,0,0,0,40,40,0,0,57,57,0,0,0,0,40,57,0,0],[75,0,0,0,0,0,0,75,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,40,57,0,0,0,0,57,57] >;`

D248C4 in GAP, Magma, Sage, TeX

`D_{24}\rtimes_8C_4`
`% in TeX`

`G:=Group("D24:8C4");`
`// GroupNames label`

`G:=SmallGroup(192,47);`
`// by ID`

`G=gap.SmallGroup(192,47);`
`# by ID`

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

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

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