Copied to
clipboard

## G = C6.16D24order 288 = 25·32

### 5th non-split extension by C6 of D24 acting via D24/D12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C6.16D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×D12 — C6.16D24
 Lower central C32 — C3×C6 — C3×C12 — C6.16D24
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.16D24
G = < a,b,c | a6=b24=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 458 in 113 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C3⋊C8, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C3⋊Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, C4⋊Dic3, C2×C24, C2×D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C2.D24, D4⋊Dic3, C6×C3⋊C8, C12⋊Dic3, C6×D12, C6.16D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C4×S3, D12, C2×Dic3, C3⋊D4, D4⋊C4, S32, C24⋊C2, D24, D6⋊C4, D4⋊S3, D4.S3, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C2.D24, D4⋊Dic3, C3⋊D24, D12.S3, D6⋊Dic3, C6.16D24

Smallest permutation representation of C6.16D24
On 96 points
Generators in S96
```(1 71 9 55 17 63)(2 64 18 56 10 72)(3 49 11 57 19 65)(4 66 20 58 12 50)(5 51 13 59 21 67)(6 68 22 60 14 52)(7 53 15 61 23 69)(8 70 24 62 16 54)(25 90 41 82 33 74)(26 75 34 83 42 91)(27 92 43 84 35 76)(28 77 36 85 44 93)(29 94 45 86 37 78)(30 79 38 87 46 95)(31 96 47 88 39 80)(32 81 40 89 48 73)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 44 55 77)(2 76 56 43)(3 42 57 75)(4 74 58 41)(5 40 59 73)(6 96 60 39)(7 38 61 95)(8 94 62 37)(9 36 63 93)(10 92 64 35)(11 34 65 91)(12 90 66 33)(13 32 67 89)(14 88 68 31)(15 30 69 87)(16 86 70 29)(17 28 71 85)(18 84 72 27)(19 26 49 83)(20 82 50 25)(21 48 51 81)(22 80 52 47)(23 46 53 79)(24 78 54 45)```

`G:=sub<Sym(96)| (1,71,9,55,17,63)(2,64,18,56,10,72)(3,49,11,57,19,65)(4,66,20,58,12,50)(5,51,13,59,21,67)(6,68,22,60,14,52)(7,53,15,61,23,69)(8,70,24,62,16,54)(25,90,41,82,33,74)(26,75,34,83,42,91)(27,92,43,84,35,76)(28,77,36,85,44,93)(29,94,45,86,37,78)(30,79,38,87,46,95)(31,96,47,88,39,80)(32,81,40,89,48,73), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,55,77)(2,76,56,43)(3,42,57,75)(4,74,58,41)(5,40,59,73)(6,96,60,39)(7,38,61,95)(8,94,62,37)(9,36,63,93)(10,92,64,35)(11,34,65,91)(12,90,66,33)(13,32,67,89)(14,88,68,31)(15,30,69,87)(16,86,70,29)(17,28,71,85)(18,84,72,27)(19,26,49,83)(20,82,50,25)(21,48,51,81)(22,80,52,47)(23,46,53,79)(24,78,54,45)>;`

`G:=Group( (1,71,9,55,17,63)(2,64,18,56,10,72)(3,49,11,57,19,65)(4,66,20,58,12,50)(5,51,13,59,21,67)(6,68,22,60,14,52)(7,53,15,61,23,69)(8,70,24,62,16,54)(25,90,41,82,33,74)(26,75,34,83,42,91)(27,92,43,84,35,76)(28,77,36,85,44,93)(29,94,45,86,37,78)(30,79,38,87,46,95)(31,96,47,88,39,80)(32,81,40,89,48,73), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,55,77)(2,76,56,43)(3,42,57,75)(4,74,58,41)(5,40,59,73)(6,96,60,39)(7,38,61,95)(8,94,62,37)(9,36,63,93)(10,92,64,35)(11,34,65,91)(12,90,66,33)(13,32,67,89)(14,88,68,31)(15,30,69,87)(16,86,70,29)(17,28,71,85)(18,84,72,27)(19,26,49,83)(20,82,50,25)(21,48,51,81)(22,80,52,47)(23,46,53,79)(24,78,54,45) );`

`G=PermutationGroup([[(1,71,9,55,17,63),(2,64,18,56,10,72),(3,49,11,57,19,65),(4,66,20,58,12,50),(5,51,13,59,21,67),(6,68,22,60,14,52),(7,53,15,61,23,69),(8,70,24,62,16,54),(25,90,41,82,33,74),(26,75,34,83,42,91),(27,92,43,84,35,76),(28,77,36,85,44,93),(29,94,45,86,37,78),(30,79,38,87,46,95),(31,96,47,88,39,80),(32,81,40,89,48,73)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,44,55,77),(2,76,56,43),(3,42,57,75),(4,74,58,41),(5,40,59,73),(6,96,60,39),(7,38,61,95),(8,94,62,37),(9,36,63,93),(10,92,64,35),(11,34,65,91),(12,90,66,33),(13,32,67,89),(14,88,68,31),(15,30,69,87),(16,86,70,29),(17,28,71,85),(18,84,72,27),(19,26,49,83),(20,82,50,25),(21,48,51,81),(22,80,52,47),(23,46,53,79),(24,78,54,45)]])`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 12 12 2 2 4 2 2 36 36 2 ··· 2 4 4 4 12 12 12 12 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - - + + - image C1 C2 C2 C2 C4 S3 S3 D4 D4 Dic3 D6 D8 SD16 C4×S3 C3⋊D4 D12 C3⋊D4 C24⋊C2 D24 S32 D4⋊S3 D4.S3 S3×Dic3 D6⋊S3 C3⋊D12 C3⋊D24 D12.S3 kernel C6.16D24 C6×C3⋊C8 C12⋊Dic3 C6×D12 C3×D12 C2×C3⋊C8 C2×D12 C3×C12 C62 D12 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C2×C6 C6 C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 2 2 4 4 1 1 1 1 1 1 2 2

Matrix representation of C6.16D24 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 10 35 0 0 0 0 0 22 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 43 0 0 0 0 30 30
,
 66 20 0 0 0 0 56 7 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 46 27 0 0 0 0 0 27

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,35,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30],[66,56,0,0,0,0,20,7,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27] >;`

C6.16D24 in GAP, Magma, Sage, TeX

`C_6._{16}D_{24}`
`% in TeX`

`G:=Group("C6.16D24");`
`// GroupNames label`

`G:=SmallGroup(288,211);`
`// by ID`

`G=gap.SmallGroup(288,211);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽