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## G = C15⋊D12order 360 = 23·32·5

### 1st semidirect product of C15 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C15⋊D12
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C32×Dic5 — C15⋊D12
 Lower central C3×C15 — C3×C30 — C15⋊D12
 Upper central C1 — C2

Generators and relations for C15⋊D12
G = < a,b,c | a15=b12=c2=1, bab-1=a4, cac=a-1, cbc=b-1 >

Subgroups: 720 in 96 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], C5, S3 [×8], C6 [×4], D4, C32, D5, C10, C10, C12 [×4], D6 [×8], C15 [×4], C3⋊S3 [×2], C3×C6, Dic5, D10, C2×C10, D12 [×4], C5×S3 [×4], D15 [×4], C30 [×4], C3×C12, C2×C3⋊S3, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5 [×4], S3×C10 [×4], D30 [×4], C12⋊S3, C5×C3⋊S3, C3⋊D15, C3×C30, C5⋊D12 [×4], C32×Dic5, C10×C3⋊S3, C2×C3⋊D15, C15⋊D12
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D5, D6 [×4], C3⋊S3, D10, D12 [×4], C2×C3⋊S3, C5⋊D4, S3×D5 [×4], C12⋊S3, C5⋊D12 [×4], D5×C3⋊S3, C15⋊D12

Smallest permutation representation of C15⋊D12
On 180 points
Generators in S180
```(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)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)(166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
(1 131 88 145 16 115 50 180 32 99 65 155)(2 135 89 149 17 119 51 169 33 103 66 159)(3 124 90 138 18 108 52 173 34 92 67 163)(4 128 76 142 19 112 53 177 35 96 68 152)(5 132 77 146 20 116 54 166 36 100 69 156)(6 121 78 150 21 120 55 170 37 104 70 160)(7 125 79 139 22 109 56 174 38 93 71 164)(8 129 80 143 23 113 57 178 39 97 72 153)(9 133 81 147 24 117 58 167 40 101 73 157)(10 122 82 136 25 106 59 171 41 105 74 161)(11 126 83 140 26 110 60 175 42 94 75 165)(12 130 84 144 27 114 46 179 43 98 61 154)(13 134 85 148 28 118 47 168 44 102 62 158)(14 123 86 137 29 107 48 172 45 91 63 162)(15 127 87 141 30 111 49 176 31 95 64 151)
(1 32)(2 31)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(46 77)(47 76)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(91 163)(92 162)(93 161)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)(101 153)(102 152)(103 151)(104 165)(105 164)(106 139)(107 138)(108 137)(109 136)(110 150)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 140)(121 175)(122 174)(123 173)(124 172)(125 171)(126 170)(127 169)(128 168)(129 167)(130 166)(131 180)(132 179)(133 178)(134 177)(135 176)```

`G:=sub<Sym(180)| (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,131,88,145,16,115,50,180,32,99,65,155)(2,135,89,149,17,119,51,169,33,103,66,159)(3,124,90,138,18,108,52,173,34,92,67,163)(4,128,76,142,19,112,53,177,35,96,68,152)(5,132,77,146,20,116,54,166,36,100,69,156)(6,121,78,150,21,120,55,170,37,104,70,160)(7,125,79,139,22,109,56,174,38,93,71,164)(8,129,80,143,23,113,57,178,39,97,72,153)(9,133,81,147,24,117,58,167,40,101,73,157)(10,122,82,136,25,106,59,171,41,105,74,161)(11,126,83,140,26,110,60,175,42,94,75,165)(12,130,84,144,27,114,46,179,43,98,61,154)(13,134,85,148,28,118,47,168,44,102,62,158)(14,123,86,137,29,107,48,172,45,91,63,162)(15,127,87,141,30,111,49,176,31,95,64,151), (1,32)(2,31)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,77)(47,76)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,165)(105,164)(106,139)(107,138)(108,137)(109,136)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,140)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,180)(132,179)(133,178)(134,177)(135,176)>;`

`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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,131,88,145,16,115,50,180,32,99,65,155)(2,135,89,149,17,119,51,169,33,103,66,159)(3,124,90,138,18,108,52,173,34,92,67,163)(4,128,76,142,19,112,53,177,35,96,68,152)(5,132,77,146,20,116,54,166,36,100,69,156)(6,121,78,150,21,120,55,170,37,104,70,160)(7,125,79,139,22,109,56,174,38,93,71,164)(8,129,80,143,23,113,57,178,39,97,72,153)(9,133,81,147,24,117,58,167,40,101,73,157)(10,122,82,136,25,106,59,171,41,105,74,161)(11,126,83,140,26,110,60,175,42,94,75,165)(12,130,84,144,27,114,46,179,43,98,61,154)(13,134,85,148,28,118,47,168,44,102,62,158)(14,123,86,137,29,107,48,172,45,91,63,162)(15,127,87,141,30,111,49,176,31,95,64,151), (1,32)(2,31)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,77)(47,76)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,165)(105,164)(106,139)(107,138)(108,137)(109,136)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,140)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,180)(132,179)(133,178)(134,177)(135,176) );`

`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),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165),(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)], [(1,131,88,145,16,115,50,180,32,99,65,155),(2,135,89,149,17,119,51,169,33,103,66,159),(3,124,90,138,18,108,52,173,34,92,67,163),(4,128,76,142,19,112,53,177,35,96,68,152),(5,132,77,146,20,116,54,166,36,100,69,156),(6,121,78,150,21,120,55,170,37,104,70,160),(7,125,79,139,22,109,56,174,38,93,71,164),(8,129,80,143,23,113,57,178,39,97,72,153),(9,133,81,147,24,117,58,167,40,101,73,157),(10,122,82,136,25,106,59,171,41,105,74,161),(11,126,83,140,26,110,60,175,42,94,75,165),(12,130,84,144,27,114,46,179,43,98,61,154),(13,134,85,148,28,118,47,168,44,102,62,158),(14,123,86,137,29,107,48,172,45,91,63,162),(15,127,87,141,30,111,49,176,31,95,64,151)], [(1,32),(2,31),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(46,77),(47,76),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(91,163),(92,162),(93,161),(94,160),(95,159),(96,158),(97,157),(98,156),(99,155),(100,154),(101,153),(102,152),(103,151),(104,165),(105,164),(106,139),(107,138),(108,137),(109,136),(110,150),(111,149),(112,148),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,140),(121,175),(122,174),(123,173),(124,172),(125,171),(126,170),(127,169),(128,168),(129,167),(130,166),(131,180),(132,179),(133,178),(134,177),(135,176)])`

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 5A 5B 6A 6B 6C 6D 10A 10B 10C 10D 10E 10F 12A ··· 12H 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 3 3 4 5 5 6 6 6 6 10 10 10 10 10 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 18 90 2 2 2 2 10 2 2 2 2 2 2 2 2 18 18 18 18 10 ··· 10 4 ··· 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 D10 D12 C5⋊D4 S3×D5 C5⋊D12 kernel C15⋊D12 C32×Dic5 C10×C3⋊S3 C2×C3⋊D15 C3×Dic5 C3×C15 C2×C3⋊S3 C30 C3×C6 C15 C32 C6 C3 # reps 1 1 1 1 4 1 2 4 2 8 4 8 8

Matrix representation of C15⋊D12 in GL6(𝔽61)

 60 43 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 60 60 0 0 0 0 1 0
,
 30 44 0 0 0 0 53 31 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 43 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60

`G:=sub<GL(6,GF(61))| [60,18,0,0,0,0,43,18,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[30,53,0,0,0,0,44,31,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;`

C15⋊D12 in GAP, Magma, Sage, TeX

`C_{15}\rtimes D_{12}`
`% in TeX`

`G:=Group("C15:D12");`
`// GroupNames label`

`G:=SmallGroup(360,70);`
`// by ID`

`G=gap.SmallGroup(360,70);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^12=c^2=1,b*a*b^-1=a^4,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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