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## G = C24.28D6order 288 = 25·32

### 28th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24.28D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C12.26D6 — C24.28D6
 Lower central C32 — C3×C6 — C3×C12 — C24.28D6
 Upper central C1 — C2 — C4 — Q16

Generators and relations for C24.28D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=a7, cac=a-1, cbc=a12b5 >

Subgroups: 836 in 186 conjugacy classes, 55 normal (17 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×12], C6 [×4], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×4], C12 [×4], C12 [×8], D6 [×12], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊S3 [×3], C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×12], D12 [×16], C3×Q8 [×8], C4○D8, C3⋊Dic3, C3×C12, C3×C12 [×2], C2×C3⋊S3, C2×C3⋊S3 [×2], S3×C8 [×4], D24 [×4], Q82S3 [×8], C3×Q16 [×4], Q83S3 [×8], C324C8, C3×C24, C4×C3⋊S3, C4×C3⋊S3 [×2], C12⋊S3 [×2], C12⋊S3 [×2], Q8×C32 [×2], D24⋊C2 [×4], C8×C3⋊S3, C325D8, C3211SD16 [×2], C32×Q16, C12.26D6 [×2], C24.28D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C4○D8, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D24⋊C2 [×4], D4×C3⋊S3, C24.28D6

Smallest permutation representation of C24.28D6
On 144 points
Generators in S144
```(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 97 75 129 47 60 13 109 87 141 35 72)(2 104 76 136 48 67 14 116 88 124 36 55)(3 111 77 143 25 50 15 99 89 131 37 62)(4 118 78 126 26 57 16 106 90 138 38 69)(5 101 79 133 27 64 17 113 91 121 39 52)(6 108 80 140 28 71 18 120 92 128 40 59)(7 115 81 123 29 54 19 103 93 135 41 66)(8 98 82 130 30 61 20 110 94 142 42 49)(9 105 83 137 31 68 21 117 95 125 43 56)(10 112 84 144 32 51 22 100 96 132 44 63)(11 119 85 127 33 58 23 107 73 139 45 70)(12 102 86 134 34 65 24 114 74 122 46 53)
(1 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 144)(67 143)(68 142)(69 141)(70 140)(71 139)(72 138)(97 106)(98 105)(99 104)(100 103)(101 102)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 24A ··· 24H order 1 2 2 2 2 3 3 3 3 4 4 4 4 4 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 18 36 36 2 2 2 2 2 4 4 9 9 2 2 2 2 2 2 18 18 4 4 4 4 8 ··· 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D8 S3×D4 D24⋊C2 kernel C24.28D6 C8×C3⋊S3 C32⋊5D8 C32⋊11SD16 C32×Q16 C12.26D6 C3×Q16 C3⋊Dic3 C2×C3⋊S3 C24 C3×Q8 C32 C6 C3 # reps 1 1 1 2 1 2 4 1 1 4 8 4 4 8

Matrix representation of C24.28D6 in GL6(𝔽73)

 0 1 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 57 0 0 0 0 16 16
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 6 6 0 0 0 0 6 67
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 57 16 0 0 0 0 16 16

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,6,6,0,0,0,0,6,67],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,57,16,0,0,0,0,16,16] >;`

C24.28D6 in GAP, Magma, Sage, TeX

`C_{24}._{28}D_6`
`% in TeX`

`G:=Group("C24.28D6");`
`// GroupNames label`

`G:=SmallGroup(288,776);`
`// by ID`

`G=gap.SmallGroup(288,776);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=a^12*b^5>;`
`// generators/relations`

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