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G = C24.49D6order 288 = 25·32

2nd non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C24.49D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×Dic12 — C24.49D6
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C24.49D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C24.49D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=cac=a-1, cbc=a15b5 >

Subgroups: 434 in 63 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3 [×4], C6 [×2], C6, C8, D4, Q8, C32, Dic3, C12 [×2], C12 [×2], D6 [×4], C16, D8, Q16, C3⋊S3, C3×C6, C24 [×2], C24, Dic6, D12 [×4], C3×Q8, SD32, C3×Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, D24 [×3], Dic12, C3×Q16, C3×C24, C3×Dic6, C12⋊S3, C48⋊C2, C8.6D6, C3×C3⋊C16, C3×Dic12, C325D8, C24.49D6
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, D12, C3⋊D4, SD32, S32, D24, D4⋊S3, C3⋊D12, C48⋊C2, C8.6D6, C3⋊D24, C24.49D6

Smallest permutation representation of C24.49D6
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 39 21 43 17 47 13 27 9 31 5 35)(2 38 22 42 18 46 14 26 10 30 6 34)(3 37 23 41 19 45 15 25 11 29 7 33)(4 36 24 40 20 44 16 48 12 28 8 32)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 24)(20 23)(21 22)(25 29)(26 28)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 3 3 3 4 4 6 6 6 8 8 12 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 72 2 2 4 2 24 2 2 4 2 2 2 2 4 4 4 24 24 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D8 D12 C3⋊D4 SD32 D24 C48⋊C2 S32 D4⋊S3 C3⋊D12 C8.6D6 C3⋊D24 C24.49D6 kernel C24.49D6 C3×C3⋊C16 C3×Dic12 C32⋊5D8 C3⋊C16 Dic12 C3×C12 C24 C3×C6 C12 C12 C32 C6 C3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 2 4 4 8 1 1 1 2 2 4

Matrix representation of C24.49D6 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 81 95 0 0 2 79
,
 0 96 0 0 1 1 0 0 0 0 2 43 0 0 45 95
,
 0 1 0 0 1 0 0 0 0 0 18 95 0 0 16 79
`G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,81,2,0,0,95,79],[0,1,0,0,96,1,0,0,0,0,2,45,0,0,43,95],[0,1,0,0,1,0,0,0,0,0,18,16,0,0,95,79] >;`

C24.49D6 in GAP, Magma, Sage, TeX

`C_{24}._{49}D_6`
`% in TeX`

`G:=Group("C24.49D6");`
`// GroupNames label`

`G:=SmallGroup(288,197);`
`// by ID`

`G=gap.SmallGroup(288,197);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,590,58,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=c*a*c=a^-1,c*b*c=a^15*b^5>;`
`// generators/relations`

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