metabelian, supersoluble, monomial, A-group
Aliases: C15⋊1D15, C152⋊1C2, C5⋊(C3⋊D15), C3⋊(C5⋊D15), (C5×C15)⋊3S3, (C3×C15)⋊1D5, C52⋊3(C3⋊S3), C32⋊2(C5⋊D5), SmallGroup(450,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C152 — C15⋊D15 |
Generators and relations for C15⋊D15
G = < a,b,c | a15=b15=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1280 in 96 conjugacy classes, 49 normal (5 characteristic)
C1, C2, C3, C5, S3, C32, D5, C15, C3⋊S3, C52, D15, C3×C15, C5⋊D5, C5×C15, C3⋊D15, C5⋊D15, C152, C15⋊D15
Quotients: C1, C2, S3, D5, C3⋊S3, D15, C5⋊D5, C3⋊D15, C5⋊D15, C15⋊D15
►On 225 points
Generators in S225
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)
(1 103 72 174 33 121 151 222 21 194 59 197 109 141 81)(2 104 73 175 34 122 152 223 22 195 60 198 110 142 82)(3 105 74 176 35 123 153 224 23 181 46 199 111 143 83)(4 91 75 177 36 124 154 225 24 182 47 200 112 144 84)(5 92 61 178 37 125 155 211 25 183 48 201 113 145 85)(6 93 62 179 38 126 156 212 26 184 49 202 114 146 86)(7 94 63 180 39 127 157 213 27 185 50 203 115 147 87)(8 95 64 166 40 128 158 214 28 186 51 204 116 148 88)(9 96 65 167 41 129 159 215 29 187 52 205 117 149 89)(10 97 66 168 42 130 160 216 30 188 53 206 118 150 90)(11 98 67 169 43 131 161 217 16 189 54 207 119 136 76)(12 99 68 170 44 132 162 218 17 190 55 208 120 137 77)(13 100 69 171 45 133 163 219 18 191 56 209 106 138 78)(14 101 70 172 31 134 164 220 19 192 57 210 107 139 79)(15 102 71 173 32 135 165 221 20 193 58 196 108 140 80)
(1 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 156)(17 155)(18 154)(19 153)(20 152)(21 151)(22 165)(23 164)(24 163)(25 162)(26 161)(27 160)(28 159)(29 158)(30 157)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(91 138)(92 137)(93 136)(94 150)(95 149)(96 148)(97 147)(98 146)(99 145)(100 144)(101 143)(102 142)(103 141)(104 140)(105 139)(121 194)(122 193)(123 192)(124 191)(125 190)(126 189)(127 188)(128 187)(129 186)(130 185)(131 184)(132 183)(133 182)(134 181)(135 195)(166 205)(167 204)(168 203)(169 202)(170 201)(171 200)(172 199)(173 198)(174 197)(175 196)(176 210)(177 209)(178 208)(179 207)(180 206)(211 218)(212 217)(213 216)(214 215)(219 225)(220 224)(221 223)
G:=sub<Sym(225)| (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (1,103,72,174,33,121,151,222,21,194,59,197,109,141,81)(2,104,73,175,34,122,152,223,22,195,60,198,110,142,82)(3,105,74,176,35,123,153,224,23,181,46,199,111,143,83)(4,91,75,177,36,124,154,225,24,182,47,200,112,144,84)(5,92,61,178,37,125,155,211,25,183,48,201,113,145,85)(6,93,62,179,38,126,156,212,26,184,49,202,114,146,86)(7,94,63,180,39,127,157,213,27,185,50,203,115,147,87)(8,95,64,166,40,128,158,214,28,186,51,204,116,148,88)(9,96,65,167,41,129,159,215,29,187,52,205,117,149,89)(10,97,66,168,42,130,160,216,30,188,53,206,118,150,90)(11,98,67,169,43,131,161,217,16,189,54,207,119,136,76)(12,99,68,170,44,132,162,218,17,190,55,208,120,137,77)(13,100,69,171,45,133,163,219,18,191,56,209,106,138,78)(14,101,70,172,31,134,164,220,19,192,57,210,107,139,79)(15,102,71,173,32,135,165,221,20,193,58,196,108,140,80), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,156)(17,155)(18,154)(19,153)(20,152)(21,151)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(91,138)(92,137)(93,136)(94,150)(95,149)(96,148)(97,147)(98,146)(99,145)(100,144)(101,143)(102,142)(103,141)(104,140)(105,139)(121,194)(122,193)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(133,182)(134,181)(135,195)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,210)(177,209)(178,208)(179,207)(180,206)(211,218)(212,217)(213,216)(214,215)(219,225)(220,224)(221,223)>;
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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (1,103,72,174,33,121,151,222,21,194,59,197,109,141,81)(2,104,73,175,34,122,152,223,22,195,60,198,110,142,82)(3,105,74,176,35,123,153,224,23,181,46,199,111,143,83)(4,91,75,177,36,124,154,225,24,182,47,200,112,144,84)(5,92,61,178,37,125,155,211,25,183,48,201,113,145,85)(6,93,62,179,38,126,156,212,26,184,49,202,114,146,86)(7,94,63,180,39,127,157,213,27,185,50,203,115,147,87)(8,95,64,166,40,128,158,214,28,186,51,204,116,148,88)(9,96,65,167,41,129,159,215,29,187,52,205,117,149,89)(10,97,66,168,42,130,160,216,30,188,53,206,118,150,90)(11,98,67,169,43,131,161,217,16,189,54,207,119,136,76)(12,99,68,170,44,132,162,218,17,190,55,208,120,137,77)(13,100,69,171,45,133,163,219,18,191,56,209,106,138,78)(14,101,70,172,31,134,164,220,19,192,57,210,107,139,79)(15,102,71,173,32,135,165,221,20,193,58,196,108,140,80), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,156)(17,155)(18,154)(19,153)(20,152)(21,151)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(91,138)(92,137)(93,136)(94,150)(95,149)(96,148)(97,147)(98,146)(99,145)(100,144)(101,143)(102,142)(103,141)(104,140)(105,139)(121,194)(122,193)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(133,182)(134,181)(135,195)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,210)(177,209)(178,208)(179,207)(180,206)(211,218)(212,217)(213,216)(214,215)(219,225)(220,224)(221,223) );
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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)], [(1,103,72,174,33,121,151,222,21,194,59,197,109,141,81),(2,104,73,175,34,122,152,223,22,195,60,198,110,142,82),(3,105,74,176,35,123,153,224,23,181,46,199,111,143,83),(4,91,75,177,36,124,154,225,24,182,47,200,112,144,84),(5,92,61,178,37,125,155,211,25,183,48,201,113,145,85),(6,93,62,179,38,126,156,212,26,184,49,202,114,146,86),(7,94,63,180,39,127,157,213,27,185,50,203,115,147,87),(8,95,64,166,40,128,158,214,28,186,51,204,116,148,88),(9,96,65,167,41,129,159,215,29,187,52,205,117,149,89),(10,97,66,168,42,130,160,216,30,188,53,206,118,150,90),(11,98,67,169,43,131,161,217,16,189,54,207,119,136,76),(12,99,68,170,44,132,162,218,17,190,55,208,120,137,77),(13,100,69,171,45,133,163,219,18,191,56,209,106,138,78),(14,101,70,172,31,134,164,220,19,192,57,210,107,139,79),(15,102,71,173,32,135,165,221,20,193,58,196,108,140,80)], [(1,81),(2,80),(3,79),(4,78),(5,77),(6,76),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,156),(17,155),(18,154),(19,153),(20,152),(21,151),(22,165),(23,164),(24,163),(25,162),(26,161),(27,160),(28,159),(29,158),(30,157),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(91,138),(92,137),(93,136),(94,150),(95,149),(96,148),(97,147),(98,146),(99,145),(100,144),(101,143),(102,142),(103,141),(104,140),(105,139),(121,194),(122,193),(123,192),(124,191),(125,190),(126,189),(127,188),(128,187),(129,186),(130,185),(131,184),(132,183),(133,182),(134,181),(135,195),(166,205),(167,204),(168,203),(169,202),(170,201),(171,200),(172,199),(173,198),(174,197),(175,196),(176,210),(177,209),(178,208),(179,207),(180,206),(211,218),(212,217),(213,216),(214,215),(219,225),(220,224),(221,223)]])
114 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 5A | ··· | 5L | 15A | ··· | 15CR |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 5 | ··· | 5 | 15 | ··· | 15 |
| size | 1 | 225 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D5 | D15 |
| kernel | C15⋊D15 | C152 | C5×C15 | C3×C15 | C15 |
| # reps | 1 | 1 | 4 | 12 | 96 |
Matrix representation of C15⋊D15 ►in GL4(𝔽31) generated by
| 11 | 26 | 0 | 0 |
| 8 | 19 | 0 | 0 |
| 0 | 0 | 4 | 12 |
| 0 | 0 | 19 | 3 |
| 3 | 11 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 30 | 18 |
| 0 | 0 | 13 | 13 |
| 11 | 26 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 13 | 1 |
G:=sub<GL(4,GF(31))| [11,8,0,0,26,19,0,0,0,0,4,19,0,0,12,3],[3,1,0,0,11,4,0,0,0,0,30,13,0,0,18,13],[11,24,0,0,26,20,0,0,0,0,30,13,0,0,0,1] >;
C15⋊D15 in GAP, Magma, Sage, TeX
C_{15}\rtimes D_{15} % in TeX
G:=Group("C15:D15"); // GroupNames label
G:=SmallGroup(450,33);
// by ID
G=gap.SmallGroup(450,33);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,41,182,1443,9004]);
// Polycyclic
G:=Group<a,b,c|a^15=b^15=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations