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## G = C24⋊4D6order 288 = 25·32

### 4th semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊4D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — C24⋊4D6
 Lower central C32 — C3×C6 — C3×C12 — C24⋊4D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C244D6
G = < a,b,c | a24=b6=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >

Subgroups: 882 in 163 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×10], C6 [×2], C6 [×5], C8, C8, C2×C4, D4 [×6], C23 [×2], C32, Dic3 [×3], C12 [×2], C12, D6 [×15], C2×C6 [×4], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, C4×S3 [×3], D12 [×4], C3⋊D4 [×4], C3×D4 [×4], C22×S3 [×4], C2×D8, C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×4], C2×C3⋊S3, S3×C8 [×3], D24 [×2], D4⋊S3 [×4], C3×D8 [×2], S3×D4 [×4], C324C8, C3×C24, D6⋊S3 [×2], C3×D12 [×4], C4×C3⋊S3, C2×S32 [×2], S3×D8 [×2], C322D8 [×2], C3×D24 [×2], C8×C3⋊S3, D6⋊D6 [×2], C244D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, C22×S3 [×2], C2×D8, S32, S3×D4 [×2], C2×S32, S3×D8 [×2], D6⋊D6, C244D6

Smallest permutation representation of C244D6
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 41 17 25 9 33)(2 40 18 48 10 32)(3 39 19 47 11 31)(4 38 20 46 12 30)(5 37 21 45 13 29)(6 36 22 44 14 28)(7 35 23 43 15 27)(8 34 24 42 16 26)
(1 5)(2 22)(3 15)(4 8)(6 18)(7 11)(9 21)(10 14)(12 24)(13 17)(16 20)(19 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 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), (1,41,17,25,9,33)(2,40,18,48,10,32)(3,39,19,47,11,31)(4,38,20,46,12,30)(5,37,21,45,13,29)(6,36,22,44,14,28)(7,35,23,43,15,27)(8,34,24,42,16,26), (1,5)(2,22)(3,15)(4,8)(6,18)(7,11)(9,21)(10,14)(12,24)(13,17)(16,20)(19,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)>;`

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 9 9 12 12 12 12 2 2 4 2 18 2 2 4 24 24 24 24 2 2 18 18 4 4 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 S32 S3×D4 C2×S32 S3×D8 D6⋊D6 C24⋊4D6 kernel C24⋊4D6 C32⋊2D8 C3×D24 C8×C3⋊S3 D6⋊D6 D24 C3⋊Dic3 C2×C3⋊S3 C24 D12 C3⋊S3 C8 C6 C4 C3 C2 C1 # reps 1 2 2 1 2 2 1 1 2 4 4 1 2 1 4 2 4

Matrix representation of C244D6 in GL6(𝔽73)

 16 57 0 0 0 0 16 16 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 14 43 0 0 0 0 43 59 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[14,43,0,0,0,0,43,59,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

C244D6 in GAP, Magma, Sage, TeX

`C_{24}\rtimes_4D_6`
`% in TeX`

`G:=Group("C24:4D6");`
`// GroupNames label`

`G:=SmallGroup(288,445);`
`// by ID`

`G=gap.SmallGroup(288,445);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,142,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;`
`// generators/relations`

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