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## G = C24.23D6order 288 = 25·32

### 23rd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24.23D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — C24.23D6
 Lower central C32 — C3×C6 — C3×C12 — C24.23D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C24.23D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=a-1, cac=a17, cbc=b5 >

Subgroups: 498 in 131 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×6], C6 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×6], C32, Dic3 [×7], C12 [×2], C12 [×5], D6 [×3], C2×C8, Q16 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×4], Dic6 [×4], C4×S3 [×7], C3×Q8 [×4], C2×Q16, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8 [×3], Dic12 [×2], C3⋊Q16 [×4], C3×Q16 [×2], S3×Q8 [×4], C324C8, C3×C24, C6.D6 [×2], C322Q8 [×2], C3×Dic6 [×4], C4×C3⋊S3, S3×Q16 [×2], C322Q16 [×2], C3×Dic12 [×2], C8×C3⋊S3, Dic3.D6 [×2], C24.23D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, C22×S3 [×2], C2×Q16, S32, S3×D4 [×2], C2×S32, S3×Q16 [×2], D6⋊D6, C24.23D6

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 9 9 2 2 4 2 12 12 12 12 18 2 2 4 2 2 18 18 4 4 4 4 24 24 24 24 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 Q16 S32 S3×D4 C2×S32 S3×Q16 D6⋊D6 C24.23D6 kernel C24.23D6 C32⋊2Q16 C3×Dic12 C8×C3⋊S3 Dic3.D6 Dic12 C3⋊Dic3 C2×C3⋊S3 C24 Dic6 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 C24.23D6 in GL6(𝔽73)

 0 41 0 0 0 0 16 41 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
,
 2 31 0 0 0 0 54 71 0 0 0 0 0 0 0 72 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[2,54,0,0,0,0,31,71,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

C24.23D6 in GAP, Magma, Sage, TeX

`C_{24}._{23}D_6`
`% in TeX`

`G:=Group("C24.23D6");`
`// GroupNames label`

`G:=SmallGroup(288,450);`
`// by ID`

`G=gap.SmallGroup(288,450);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,142,675,346,80,1356,9414]);`
`// Polycyclic`

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

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