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## G = C24⋊6D6order 288 = 25·32

### 6th semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊6D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — C24⋊6D6
 Lower central C32 — C3×C6 — C3×C12 — C24⋊6D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C246D6
G = < a,b,c | a24=b6=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 722 in 146 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×6], C6 [×2], C6 [×4], C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3 [×4], C12 [×2], C12 [×2], D6 [×10], C2×C6 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6, C4×S3 [×4], D12 [×3], D12, C2×Dic3, C3⋊D4 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×3], C2×C3⋊S3, C8⋊S3 [×3], C24⋊C2, D24, D4⋊S3 [×2], D4.S3, Q82S3, C3×D8, C3×SD16, S3×D4 [×2], D42S3, Q83S3, C324C8, C3×C24, S3×Dic3, D6⋊S3, C3⋊D12, C3×Dic6, C3×D12 [×3], C4×C3⋊S3, C2×S32, D8⋊S3, Q83D6, C322D8, Dic6⋊S3, C3×C24⋊C2, C3×D24, C24⋊S3, D12⋊S3, D6⋊D6, C246D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, D8⋊S3, Q83D6, D6⋊D6, C246D6

Smallest permutation representation of C246D6
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 17)(2 22)(4 8)(5 13)(6 18)(7 23)(10 14)(11 19)(12 24)(16 20)(25 33)(26 38)(27 43)(28 48)(30 34)(31 39)(32 44)(36 40)(37 45)(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), (1,17)(2,22)(4,8)(5,13)(6,18)(7,23)(10,14)(11,19)(12,24)(16,20)(25,33)(26,38)(27,43)(28,48)(30,34)(31,39)(32,44)(36,40)(37,45)(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), (1,17)(2,22)(4,8)(5,13)(6,18)(7,23)(10,14)(11,19)(12,24)(16,20)(25,33)(26,38)(27,43)(28,48)(30,34)(31,39)(32,44)(36,40)(37,45)(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)], [(1,17),(2,22),(4,8),(5,13),(6,18),(7,23),(10,14),(11,19),(12,24),(16,20),(25,33),(26,38),(27,43),(28,48),(30,34),(31,39),(32,44),(36,40),(37,45),(42,46)])`

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 6 6 6 8 8 12 12 12 12 12 24 ··· 24 size 1 1 12 12 12 18 2 2 4 2 12 18 2 2 4 24 24 24 4 36 4 4 4 4 24 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C8⋊C22 S32 S3×D4 C2×S32 D8⋊S3 Q8⋊3D6 D6⋊D6 C24⋊6D6 kernel C24⋊6D6 C32⋊2D8 Dic6⋊S3 C3×C24⋊C2 C3×D24 C24⋊S3 D12⋊S3 D6⋊D6 C24⋊C2 D24 C3⋊Dic3 C2×C3⋊S3 C24 Dic6 D12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 1 3 1 1 2 1 2 2 2 4

Matrix representation of C246D6 in GL4(𝔽5) generated by

 0 0 3 0 0 4 0 1 1 0 1 0 0 3 0 4
,
 0 1 0 1 0 0 3 0 0 4 0 2 1 0 3 0
,
 1 0 0 0 0 4 0 0 2 0 4 0 0 0 0 1
`G:=sub<GL(4,GF(5))| [0,0,1,0,0,4,0,3,3,0,1,0,0,1,0,4],[0,0,0,1,1,0,4,0,0,3,0,3,1,0,2,0],[1,0,2,0,0,4,0,0,0,0,4,0,0,0,0,1] >;`

C246D6 in GAP, Magma, Sage, TeX

`C_{24}\rtimes_6D_6`
`% in TeX`

`G:=Group("C24:6D6");`
`// GroupNames label`

`G:=SmallGroup(288,446);`
`// by ID`

`G=gap.SmallGroup(288,446);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,58,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;`
`// generators/relations`

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