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## G = C24.100D4order 192 = 26·3

### 23rd non-split extension by C24 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C24.100D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.13D6 — C24.100D4
 Lower central C3 — C6 — C12 — C24.100D4
 Upper central C1 — C8 — C2×C8 — C8○D4

Generators and relations for C24.100D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=cac=a17, cbc=a12b3 >

Subgroups: 240 in 106 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, Dic3 [×3], C12 [×2], C12, D6, C2×C6, C2×C6, C42, C2×C8, C2×C8 [×3], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4×C8, C4≀C2 [×2], C8.C4, C8○D4, C8○D4, C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C4○D12, C3×C4○D4, C8○D8, C8×Dic3, C12.53D4, D12⋊C4, Q83Dic3, C8○D12, Q8.13D6, C3×C8○D4, C24.100D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C8○D8, C4×C3⋊D4, C24.100D4

Smallest permutation representation of C24.100D4
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 16 7 22 13 4 19 10)(2 9 8 15 14 21 20 3)(5 12 11 18 17 24 23 6)(25 34 43 28 37 46 31 40)(26 27 44 45 38 39 32 33)(29 30 47 48 41 42 35 36)
(1 37)(2 30)(3 47)(4 40)(5 33)(6 26)(7 43)(8 36)(9 29)(10 46)(11 39)(12 32)(13 25)(14 42)(15 35)(16 28)(17 45)(18 38)(19 31)(20 48)(21 41)(22 34)(23 27)(24 44)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 12A 12B 12C 12D 12E 24A 24B 24C 24D 24E ··· 24J order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 12 12 12 12 12 24 24 24 24 24 ··· 24 size 1 1 2 4 12 2 1 1 2 4 6 6 6 6 12 2 4 4 4 1 1 1 1 2 2 4 4 6 6 6 6 12 12 2 2 4 4 4 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4×S3 C4×S3 C4○D12 C8○D8 C24.100D4 kernel C24.100D4 C8×Dic3 C12.53D4 D12⋊C4 Q8⋊3Dic3 C8○D12 Q8.13D6 C3×C8○D4 D4⋊S3 D4.S3 Q8⋊2S3 C3⋊Q16 C8○D4 C24 C2×C8 M4(2) C4○D4 C2×C6 C8 D4 Q8 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 2 4 2 2 4 8 4

Matrix representation of C24.100D4 in GL4(𝔽73) generated by

 1 1 0 0 72 0 0 0 0 0 22 0 0 0 0 22
,
 72 0 0 0 1 1 0 0 0 0 10 0 0 0 0 22
,
 1 0 0 0 72 72 0 0 0 0 0 22 0 0 10 0
`G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,22,0,0,0,0,22],[72,1,0,0,0,1,0,0,0,0,10,0,0,0,0,22],[1,72,0,0,0,72,0,0,0,0,0,10,0,0,22,0] >;`

C24.100D4 in GAP, Magma, Sage, TeX

`C_{24}._{100}D_4`
`% in TeX`

`G:=Group("C24.100D4");`
`// GroupNames label`

`G:=SmallGroup(192,703);`
`// by ID`

`G=gap.SmallGroup(192,703);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,58,136,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=c*a*c=a^17,c*b*c=a^12*b^3>;`
`// generators/relations`

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