metabelian, supersoluble, monomial
Aliases: C15⋊1D12, C30.14D6, (C3×C15)⋊9D4, Dic5⋊(C3⋊S3), C6.21(S3×D5), C5⋊2(C12⋊S3), C3⋊1(C5⋊D12), (C3×Dic5)⋊3S3, (C3×C6).25D10, C32⋊6(C5⋊D4), (C3×C30).13C22, (C32×Dic5)⋊6C2, (C2×C3⋊S3)⋊2D5, C2.6(D5×C3⋊S3), (C10×C3⋊S3)⋊2C2, (C2×C3⋊D15)⋊4C2, C10.6(C2×C3⋊S3), SmallGroup(360,70)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C22, C5, S3, C6, D4, C32, D5, C10, C10, C12, D6, C15, C3⋊S3, C3×C6, Dic5, D10, C2×C10, D12, C5×S3, D15, C30, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5, S3×C10, D30, C12⋊S3, C5×C3⋊S3, C3⋊D15, C3×C30, C5⋊D12, C32×Dic5, C10×C3⋊S3, C2×C3⋊D15, C15⋊D12
Quotients: C1, C2, C22, S3, D4, D5, D6, C3⋊S3, D10, D12, C2×C3⋊S3, C5⋊D4, S3×D5, C12⋊S3, C5⋊D12, D5×C3⋊S3, 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)
(1 124 84 142 20 110 46 172 35 102 65 162)(2 128 85 146 21 114 47 176 36 91 66 151)(3 132 86 150 22 118 48 180 37 95 67 155)(4 121 87 139 23 107 49 169 38 99 68 159)(5 125 88 143 24 111 50 173 39 103 69 163)(6 129 89 147 25 115 51 177 40 92 70 152)(7 133 90 136 26 119 52 166 41 96 71 156)(8 122 76 140 27 108 53 170 42 100 72 160)(9 126 77 144 28 112 54 174 43 104 73 164)(10 130 78 148 29 116 55 178 44 93 74 153)(11 134 79 137 30 120 56 167 45 97 75 157)(12 123 80 141 16 109 57 171 31 101 61 161)(13 127 81 145 17 113 58 175 32 105 62 165)(14 131 82 149 18 117 59 179 33 94 63 154)(15 135 83 138 19 106 60 168 34 98 64 158)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(91 158)(92 157)(93 156)(94 155)(95 154)(96 153)(97 152)(98 151)(99 165)(100 164)(101 163)(102 162)(103 161)(104 160)(105 159)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 150)(118 149)(119 148)(120 147)(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), (1,124,84,142,20,110,46,172,35,102,65,162)(2,128,85,146,21,114,47,176,36,91,66,151)(3,132,86,150,22,118,48,180,37,95,67,155)(4,121,87,139,23,107,49,169,38,99,68,159)(5,125,88,143,24,111,50,173,39,103,69,163)(6,129,89,147,25,115,51,177,40,92,70,152)(7,133,90,136,26,119,52,166,41,96,71,156)(8,122,76,140,27,108,53,170,42,100,72,160)(9,126,77,144,28,112,54,174,43,104,73,164)(10,130,78,148,29,116,55,178,44,93,74,153)(11,134,79,137,30,120,56,167,45,97,75,157)(12,123,80,141,16,109,57,171,31,101,61,161)(13,127,81,145,17,113,58,175,32,105,62,165)(14,131,82,149,18,117,59,179,33,94,63,154)(15,135,83,138,19,106,60,168,34,98,64,158), (1,35)(2,34)(3,33)(4,32)(5,31)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,165)(100,164)(101,163)(102,162)(103,161)(104,160)(105,159)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,150)(118,149)(119,148)(120,147)(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), (1,124,84,142,20,110,46,172,35,102,65,162)(2,128,85,146,21,114,47,176,36,91,66,151)(3,132,86,150,22,118,48,180,37,95,67,155)(4,121,87,139,23,107,49,169,38,99,68,159)(5,125,88,143,24,111,50,173,39,103,69,163)(6,129,89,147,25,115,51,177,40,92,70,152)(7,133,90,136,26,119,52,166,41,96,71,156)(8,122,76,140,27,108,53,170,42,100,72,160)(9,126,77,144,28,112,54,174,43,104,73,164)(10,130,78,148,29,116,55,178,44,93,74,153)(11,134,79,137,30,120,56,167,45,97,75,157)(12,123,80,141,16,109,57,171,31,101,61,161)(13,127,81,145,17,113,58,175,32,105,62,165)(14,131,82,149,18,117,59,179,33,94,63,154)(15,135,83,138,19,106,60,168,34,98,64,158), (1,35)(2,34)(3,33)(4,32)(5,31)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,165)(100,164)(101,163)(102,162)(103,161)(104,160)(105,159)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,150)(118,149)(119,148)(120,147)(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)], [(1,124,84,142,20,110,46,172,35,102,65,162),(2,128,85,146,21,114,47,176,36,91,66,151),(3,132,86,150,22,118,48,180,37,95,67,155),(4,121,87,139,23,107,49,169,38,99,68,159),(5,125,88,143,24,111,50,173,39,103,69,163),(6,129,89,147,25,115,51,177,40,92,70,152),(7,133,90,136,26,119,52,166,41,96,71,156),(8,122,76,140,27,108,53,170,42,100,72,160),(9,126,77,144,28,112,54,174,43,104,73,164),(10,130,78,148,29,116,55,178,44,93,74,153),(11,134,79,137,30,120,56,167,45,97,75,157),(12,123,80,141,16,109,57,171,31,101,61,161),(13,127,81,145,17,113,58,175,32,105,62,165),(14,131,82,149,18,117,59,179,33,94,63,154),(15,135,83,138,19,106,60,168,34,98,64,158)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,24),(17,23),(18,22),(19,21),(25,30),(26,29),(27,28),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(91,158),(92,157),(93,156),(94,155),(95,154),(96,153),(97,152),(98,151),(99,165),(100,164),(101,163),(102,162),(103,161),(104,160),(105,159),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,150),(118,149),(119,148),(120,147),(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