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## G = C24.D6order 288 = 25·32

### 46th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C24.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.D6 — C24.D6
 Lower central C32 — C3×C6 — C24.D6
 Upper central C1 — C4 — C8

Generators and relations for C24.D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=a5, cac-1=a17, cbc-1=a12b-1 >

Subgroups: 434 in 135 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×3], C2×C6 [×2], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×2], C3⋊C8 [×3], C24 [×2], C24 [×3], Dic6 [×2], C4×S3 [×2], C4×S3 [×3], D12 [×2], C3⋊D4 [×4], C2×C12 [×2], C8○D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×5], C8⋊S3 [×2], C8⋊S3 [×2], C2×C3⋊C8 [×2], C3×M4(2) [×2], C4○D12 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, D6⋊S3, C3⋊D12 [×2], C322Q8, S3×C12 [×2], C4×C3⋊S3, D12.C4 [×2], S3×C3⋊C8 [×2], C12.31D6, C3×C8⋊S3 [×2], C8×C3⋊S3, D6.D6, C24.D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C8○D4, S32, S3×C2×C4 [×2], C2×S32, D12.C4 [×2], C4×S32, C24.D6

Smallest permutation representation of C24.D6
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 26 17 34 9 42)(2 31 18 39 10 47)(3 36 19 44 11 28)(4 41 20 25 12 33)(5 46 21 30 13 38)(6 27 22 35 14 43)(7 32 23 40 15 48)(8 37 24 45 16 29)
(1 23 13 11)(2 16 14 4)(3 9 15 21)(5 19 17 7)(6 12 18 24)(8 22 20 10)(25 27 37 39)(26 44 38 32)(28 30 40 42)(29 47 41 35)(31 33 43 45)(34 36 46 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), (1,26,17,34,9,42)(2,31,18,39,10,47)(3,36,19,44,11,28)(4,41,20,25,12,33)(5,46,21,30,13,38)(6,27,22,35,14,43)(7,32,23,40,15,48)(8,37,24,45,16,29), (1,23,13,11)(2,16,14,4)(3,9,15,21)(5,19,17,7)(6,12,18,24)(8,22,20,10)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48)>;`

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 D6 C4×S3 C4×S3 C8○D4 S32 C2×S32 D12.C4 C4×S32 C24.D6 kernel C24.D6 S3×C3⋊C8 C12.31D6 C3×C8⋊S3 C8×C3⋊S3 D6.D6 D6⋊S3 C3⋊D12 C32⋊2Q8 C8⋊S3 C3⋊C8 C24 C4×S3 Dic3 D6 C32 C8 C4 C3 C2 C1 # reps 1 2 1 2 1 1 2 4 2 2 2 2 2 4 4 4 1 1 4 2 4

Matrix representation of C24.D6 in GL6(𝔽73)

 27 0 0 0 0 0 0 27 0 0 0 0 0 0 63 0 0 0 0 0 0 10 0 0 0 0 0 0 46 27 0 0 0 0 46 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 0 0 72 1 0 0 0 0 0 1

`G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,63,0,0,0,0,0,0,10,0,0,0,0,0,0,46,46,0,0,0,0,27,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;`

C24.D6 in GAP, Magma, Sage, TeX

`C_{24}.D_6`
`% in TeX`

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

`G:=SmallGroup(288,453);`
`// by ID`

`G=gap.SmallGroup(288,453);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,219,58,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;`
`// generators/relations`

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