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## G = C24.D4order 192 = 26·3

### 52nd non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.D4
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C12.C8 — C24.D4
 Lower central C3 — C6 — C2×C6 — C24.D4
 Upper central C1 — C4 — C2×C8 — C2×M4(2)

Generators and relations for C24.D4
G = < a,b,c | a24=1, b4=a18, c2=a9, bab-1=a5, cac-1=a17, cbc-1=a15b3 >

Subgroups: 104 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, M5(2), C2×M4(2), C3⋊C16, C2×C24, C3×M4(2), C22×C12, C23.C8, C12.C8, C6×M4(2), C24.D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C2×C3⋊C8, C4.Dic3, C6.D4, C23.C8, C12.55D4, C24.D4

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A ··· 24H order 1 2 2 2 3 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 ··· 24 size 1 1 2 4 2 1 1 2 4 2 2 2 4 4 2 2 2 2 4 4 2 2 2 2 4 4 12 ··· 12 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C4 C4 C8 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C3⋊C8 C4.Dic3 C23.C8 C24.D4 kernel C24.D4 C12.C8 C6×M4(2) C2×C24 C22×C12 C2×C12 C22×C6 C2×M4(2) C24 C2×C8 C2×C8 C22×C4 C12 C8 C2×C4 C23 C4 C3 C1 # reps 1 2 1 2 2 4 4 1 2 1 1 1 2 4 2 2 4 2 4

Matrix representation of C24.D4 in GL4(𝔽97) generated by

 61 10 0 0 65 36 0 0 0 0 35 28 0 0 85 62
,
 0 0 1 22 0 0 0 96 96 37 0 0 53 1 0 0
,
 0 0 1 0 0 0 0 1 1 59 0 0 44 96 0 0
`G:=sub<GL(4,GF(97))| [61,65,0,0,10,36,0,0,0,0,35,85,0,0,28,62],[0,0,96,53,0,0,37,1,1,0,0,0,22,96,0,0],[0,0,1,44,0,0,59,96,1,0,0,0,0,1,0,0] >;`

C24.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(192,112);`
`// by ID`

`G=gap.SmallGroup(192,112);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,100,1123,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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