Copied to
clipboard

## G = C24.62D6order 288 = 25·32

### 15th non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C24.62D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×C3⋊C16 — C24.62D6
 Lower central C32 — C3×C6 — C24.62D6
 Upper central C1 — C8

Generators and relations for C24.62D6
G = < a,b,c | a24=1, b6=a3, c2=a6, bab-1=cac-1=a17, cbc-1=a18b5 >

Subgroups: 210 in 63 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C24, C4×S3, M5(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, S3×C8, C324C8, C3×C24, C4×C3⋊S3, D6.C8, C3×C3⋊C16, C8×C3⋊S3, C24.62D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, M5(2), S32, S3×C8, C6.D6, D6.C8, C12.29D6, C24.62D6

Smallest permutation representation of C24.62D6
On 48 points
Generators in S48
```(1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47)(2 36 22 8 42 28 14 48 34 20 6 40 26 12 46 32 18 4 38 24 10 44 30 16)
(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 13 3 25 15 37 27)(2 32 14 44 26 8 38 20)(4 18 16 30 28 42 40 6)(5 11 17 23 29 35 41 47)(7 45 19 9 31 21 43 33)(10 24 22 36 34 48 46 12)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A ··· 24H 24I 24J 24K 24L 48A ··· 48P order 1 2 2 3 3 3 4 4 4 6 6 6 8 8 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 ··· 24 24 24 24 24 48 ··· 48 size 1 1 18 2 2 4 1 1 18 2 2 4 1 1 1 1 18 18 2 2 2 2 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 C8 S3 D6 C4×S3 M5(2) S3×C8 D6.C8 S32 C6.D6 C12.29D6 C24.62D6 kernel C24.62D6 C3×C3⋊C16 C8×C3⋊S3 C32⋊4C8 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊C16 C24 C12 C32 C6 C3 C8 C4 C2 C1 # reps 1 2 1 2 2 4 4 2 2 4 4 8 16 1 1 2 4

Matrix representation of C24.62D6 in GL6(𝔽97)

 0 96 0 0 0 0 1 1 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 75 0 0 0 0 0 0 75
,
 22 0 0 0 0 0 75 75 0 0 0 0 0 0 50 47 0 0 0 0 49 47 0 0 0 0 0 0 50 47 0 0 0 0 50 0
,
 96 0 0 0 0 0 1 1 0 0 0 0 0 0 64 0 0 0 0 0 31 33 0 0 0 0 0 0 0 22 0 0 0 0 22 0

`G:=sub<GL(6,GF(97))| [0,1,0,0,0,0,96,1,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[22,75,0,0,0,0,0,75,0,0,0,0,0,0,50,49,0,0,0,0,47,47,0,0,0,0,0,0,50,50,0,0,0,0,47,0],[96,1,0,0,0,0,0,1,0,0,0,0,0,0,64,31,0,0,0,0,0,33,0,0,0,0,0,0,0,22,0,0,0,0,22,0] >;`

C24.62D6 in GAP, Magma, Sage, TeX

`C_{24}._{62}D_6`
`% in TeX`

`G:=Group("C24.62D6");`
`// GroupNames label`

`G:=SmallGroup(288,192);`
`// by ID`

`G=gap.SmallGroup(288,192);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,36,58,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^6=a^3,c^2=a^6,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^18*b^5>;`
`// generators/relations`

׿
×
𝔽