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## G = C6.D24order 288 = 25·32

### 8th non-split extension by C6 of D24 acting via D24/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C6.D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊5D8 — C6.D24
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C6.D24
 Upper central C1 — C2 — C4 — C8 — C16

Generators and relations for C6.D24
G = < a,b,c | a6=1, b24=c2=a3, ab=ba, cac-1=a-1, cbc-1=b23 >

Subgroups: 536 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, D4, Q8, C32, Dic3, C12, D6, C16, D8, Q16, C3⋊S3, C3×C6, C24, Dic6, D12, SD32, C3⋊Dic3, C3×C12, C2×C3⋊S3, C48, D24, Dic12, C3×C24, C324Q8, C12⋊S3, C48⋊C2, C3×C48, C325D8, C325Q16, C6.D24
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, SD32, C2×C3⋊S3, D24, C12⋊S3, C48⋊C2, C325D8, C6.D24

Smallest permutation representation of C6.D24
On 144 points
Generators in S144
```(1 98 77 25 122 53)(2 99 78 26 123 54)(3 100 79 27 124 55)(4 101 80 28 125 56)(5 102 81 29 126 57)(6 103 82 30 127 58)(7 104 83 31 128 59)(8 105 84 32 129 60)(9 106 85 33 130 61)(10 107 86 34 131 62)(11 108 87 35 132 63)(12 109 88 36 133 64)(13 110 89 37 134 65)(14 111 90 38 135 66)(15 112 91 39 136 67)(16 113 92 40 137 68)(17 114 93 41 138 69)(18 115 94 42 139 70)(19 116 95 43 140 71)(20 117 96 44 141 72)(21 118 49 45 142 73)(22 119 50 46 143 74)(23 120 51 47 144 75)(24 121 52 48 97 76)
(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 24 25 48)(2 47 26 23)(3 22 27 46)(4 45 28 21)(5 20 29 44)(6 43 30 19)(7 18 31 42)(8 41 32 17)(9 16 33 40)(10 39 34 15)(11 14 35 38)(12 37 36 13)(49 125 73 101)(50 100 74 124)(51 123 75 99)(52 98 76 122)(53 121 77 97)(54 144 78 120)(55 119 79 143)(56 142 80 118)(57 117 81 141)(58 140 82 116)(59 115 83 139)(60 138 84 114)(61 113 85 137)(62 136 86 112)(63 111 87 135)(64 134 88 110)(65 109 89 133)(66 132 90 108)(67 107 91 131)(68 130 92 106)(69 105 93 129)(70 128 94 104)(71 103 95 127)(72 126 96 102)```

`G:=sub<Sym(144)| (1,98,77,25,122,53)(2,99,78,26,123,54)(3,100,79,27,124,55)(4,101,80,28,125,56)(5,102,81,29,126,57)(6,103,82,30,127,58)(7,104,83,31,128,59)(8,105,84,32,129,60)(9,106,85,33,130,61)(10,107,86,34,131,62)(11,108,87,35,132,63)(12,109,88,36,133,64)(13,110,89,37,134,65)(14,111,90,38,135,66)(15,112,91,39,136,67)(16,113,92,40,137,68)(17,114,93,41,138,69)(18,115,94,42,139,70)(19,116,95,43,140,71)(20,117,96,44,141,72)(21,118,49,45,142,73)(22,119,50,46,143,74)(23,120,51,47,144,75)(24,121,52,48,97,76), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,24,25,48)(2,47,26,23)(3,22,27,46)(4,45,28,21)(5,20,29,44)(6,43,30,19)(7,18,31,42)(8,41,32,17)(9,16,33,40)(10,39,34,15)(11,14,35,38)(12,37,36,13)(49,125,73,101)(50,100,74,124)(51,123,75,99)(52,98,76,122)(53,121,77,97)(54,144,78,120)(55,119,79,143)(56,142,80,118)(57,117,81,141)(58,140,82,116)(59,115,83,139)(60,138,84,114)(61,113,85,137)(62,136,86,112)(63,111,87,135)(64,134,88,110)(65,109,89,133)(66,132,90,108)(67,107,91,131)(68,130,92,106)(69,105,93,129)(70,128,94,104)(71,103,95,127)(72,126,96,102)>;`

`G:=Group( (1,98,77,25,122,53)(2,99,78,26,123,54)(3,100,79,27,124,55)(4,101,80,28,125,56)(5,102,81,29,126,57)(6,103,82,30,127,58)(7,104,83,31,128,59)(8,105,84,32,129,60)(9,106,85,33,130,61)(10,107,86,34,131,62)(11,108,87,35,132,63)(12,109,88,36,133,64)(13,110,89,37,134,65)(14,111,90,38,135,66)(15,112,91,39,136,67)(16,113,92,40,137,68)(17,114,93,41,138,69)(18,115,94,42,139,70)(19,116,95,43,140,71)(20,117,96,44,141,72)(21,118,49,45,142,73)(22,119,50,46,143,74)(23,120,51,47,144,75)(24,121,52,48,97,76), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,24,25,48)(2,47,26,23)(3,22,27,46)(4,45,28,21)(5,20,29,44)(6,43,30,19)(7,18,31,42)(8,41,32,17)(9,16,33,40)(10,39,34,15)(11,14,35,38)(12,37,36,13)(49,125,73,101)(50,100,74,124)(51,123,75,99)(52,98,76,122)(53,121,77,97)(54,144,78,120)(55,119,79,143)(56,142,80,118)(57,117,81,141)(58,140,82,116)(59,115,83,139)(60,138,84,114)(61,113,85,137)(62,136,86,112)(63,111,87,135)(64,134,88,110)(65,109,89,133)(66,132,90,108)(67,107,91,131)(68,130,92,106)(69,105,93,129)(70,128,94,104)(71,103,95,127)(72,126,96,102) );`

`G=PermutationGroup([[(1,98,77,25,122,53),(2,99,78,26,123,54),(3,100,79,27,124,55),(4,101,80,28,125,56),(5,102,81,29,126,57),(6,103,82,30,127,58),(7,104,83,31,128,59),(8,105,84,32,129,60),(9,106,85,33,130,61),(10,107,86,34,131,62),(11,108,87,35,132,63),(12,109,88,36,133,64),(13,110,89,37,134,65),(14,111,90,38,135,66),(15,112,91,39,136,67),(16,113,92,40,137,68),(17,114,93,41,138,69),(18,115,94,42,139,70),(19,116,95,43,140,71),(20,117,96,44,141,72),(21,118,49,45,142,73),(22,119,50,46,143,74),(23,120,51,47,144,75),(24,121,52,48,97,76)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,24,25,48),(2,47,26,23),(3,22,27,46),(4,45,28,21),(5,20,29,44),(6,43,30,19),(7,18,31,42),(8,41,32,17),(9,16,33,40),(10,39,34,15),(11,14,35,38),(12,37,36,13),(49,125,73,101),(50,100,74,124),(51,123,75,99),(52,98,76,122),(53,121,77,97),(54,144,78,120),(55,119,79,143),(56,142,80,118),(57,117,81,141),(58,140,82,116),(59,115,83,139),(60,138,84,114),(61,113,85,137),(62,136,86,112),(63,111,87,135),(64,134,88,110),(65,109,89,133),(66,132,90,108),(67,107,91,131),(68,130,92,106),(69,105,93,129),(70,128,94,104),(71,103,95,127),(72,126,96,102)]])`

75 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 12A ··· 12H 16A 16B 16C 16D 24A ··· 24P 48A ··· 48AF order 1 2 2 3 3 3 3 4 4 6 6 6 6 8 8 12 ··· 12 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 72 2 2 2 2 2 72 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

75 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D8 D12 SD32 D24 C48⋊C2 kernel C6.D24 C3×C48 C32⋊5D8 C32⋊5Q16 C48 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 4 1 4 2 8 4 16 32

Matrix representation of C6.D24 in GL4(𝔽97) generated by

 0 96 0 0 1 96 0 0 0 0 96 0 0 0 0 96
,
 58 68 0 0 29 29 0 0 0 0 95 52 0 0 45 43
,
 58 68 0 0 29 39 0 0 0 0 2 45 0 0 43 95
`G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,96,0,0,0,0,96],[58,29,0,0,68,29,0,0,0,0,95,45,0,0,52,43],[58,29,0,0,68,39,0,0,0,0,2,43,0,0,45,95] >;`

C6.D24 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(288,275);`
`// by ID`

`G=gap.SmallGroup(288,275);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=1,b^24=c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^23>;`
`// generators/relations`

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