metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊1M5(2), C5⋊C16⋊1S3, D6.(C5⋊C8), C10.6(S3×C8), C30.3(C2×C8), C5⋊2(D6.C8), C15⋊C16⋊3C2, Dic3.(C5⋊C8), (C4×S3).3F5, C4.28(S3×F5), C15⋊3C8.4C4, (S3×C10).2C8, (S3×C20).4C4, C20.28(C4×S3), C60.28(C2×C4), C5⋊2C8.29D6, C12.35(C2×F5), C3⋊1(C20.C8), (C5×Dic3).2C8, C6.2(C2×C5⋊C8), C2.3(S3×C5⋊C8), (C3×C5⋊C16)⋊3C2, (S3×C5⋊2C8).3C2, (C3×C5⋊2C8).29C22, SmallGroup(480,241)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊M5(2)
G = < a,b,c | a15=b16=c2=1, bab-1=a8, cac=a11, cbc=b9 >
Smallest permutation representation of C15⋊M5(2)
►On 240 points
Generators in S240
(1 39 236 209 126 138 93 181 203 52 112 156 161 74 26)(2 204 40 53 237 97 210 157 127 162 139 75 94 27 182)(3 128 205 163 41 140 54 76 238 95 98 28 211 183 158)(4 239 113 96 206 99 164 29 42 212 141 184 55 159 77)(5 43 240 213 114 142 81 185 207 56 100 160 165 78 30)(6 208 44 57 225 101 214 145 115 166 143 79 82 31 186)(7 116 193 167 45 144 58 80 226 83 102 32 215 187 146)(8 227 117 84 194 103 168 17 46 216 129 188 59 147 65)(9 47 228 217 118 130 85 189 195 60 104 148 169 66 18)(10 196 48 61 229 105 218 149 119 170 131 67 86 19 190)(11 120 197 171 33 132 62 68 230 87 106 20 219 191 150)(12 231 121 88 198 107 172 21 34 220 133 192 63 151 69)(13 35 232 221 122 134 89 177 199 64 108 152 173 70 22)(14 200 36 49 233 109 222 153 123 174 135 71 90 23 178)(15 124 201 175 37 136 50 72 234 91 110 24 223 179 154)(16 235 125 92 202 111 176 25 38 224 137 180 51 155 73)
(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)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(2 10)(4 12)(6 14)(8 16)(17 125)(18 118)(19 127)(20 120)(21 113)(22 122)(23 115)(24 124)(25 117)(26 126)(27 119)(28 128)(29 121)(30 114)(31 123)(32 116)(33 150)(34 159)(35 152)(36 145)(37 154)(38 147)(39 156)(40 149)(41 158)(42 151)(43 160)(44 153)(45 146)(46 155)(47 148)(48 157)(49 57)(51 59)(53 61)(55 63)(65 202)(66 195)(67 204)(68 197)(69 206)(70 199)(71 208)(72 201)(73 194)(74 203)(75 196)(76 205)(77 198)(78 207)(79 200)(80 193)(82 90)(84 92)(86 94)(88 96)(97 131)(98 140)(99 133)(100 142)(101 135)(102 144)(103 137)(104 130)(105 139)(106 132)(107 141)(108 134)(109 143)(110 136)(111 129)(112 138)(162 170)(164 172)(166 174)(168 176)(177 232)(178 225)(179 234)(180 227)(181 236)(182 229)(183 238)(184 231)(185 240)(186 233)(187 226)(188 235)(189 228)(190 237)(191 230)(192 239)(210 218)(212 220)(214 222)(216 224)
G:=sub<Sym(240)| (1,39,236,209,126,138,93,181,203,52,112,156,161,74,26)(2,204,40,53,237,97,210,157,127,162,139,75,94,27,182)(3,128,205,163,41,140,54,76,238,95,98,28,211,183,158)(4,239,113,96,206,99,164,29,42,212,141,184,55,159,77)(5,43,240,213,114,142,81,185,207,56,100,160,165,78,30)(6,208,44,57,225,101,214,145,115,166,143,79,82,31,186)(7,116,193,167,45,144,58,80,226,83,102,32,215,187,146)(8,227,117,84,194,103,168,17,46,216,129,188,59,147,65)(9,47,228,217,118,130,85,189,195,60,104,148,169,66,18)(10,196,48,61,229,105,218,149,119,170,131,67,86,19,190)(11,120,197,171,33,132,62,68,230,87,106,20,219,191,150)(12,231,121,88,198,107,172,21,34,220,133,192,63,151,69)(13,35,232,221,122,134,89,177,199,64,108,152,173,70,22)(14,200,36,49,233,109,222,153,123,174,135,71,90,23,178)(15,124,201,175,37,136,50,72,234,91,110,24,223,179,154)(16,235,125,92,202,111,176,25,38,224,137,180,51,155,73), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (2,10)(4,12)(6,14)(8,16)(17,125)(18,118)(19,127)(20,120)(21,113)(22,122)(23,115)(24,124)(25,117)(26,126)(27,119)(28,128)(29,121)(30,114)(31,123)(32,116)(33,150)(34,159)(35,152)(36,145)(37,154)(38,147)(39,156)(40,149)(41,158)(42,151)(43,160)(44,153)(45,146)(46,155)(47,148)(48,157)(49,57)(51,59)(53,61)(55,63)(65,202)(66,195)(67,204)(68,197)(69,206)(70,199)(71,208)(72,201)(73,194)(74,203)(75,196)(76,205)(77,198)(78,207)(79,200)(80,193)(82,90)(84,92)(86,94)(88,96)(97,131)(98,140)(99,133)(100,142)(101,135)(102,144)(103,137)(104,130)(105,139)(106,132)(107,141)(108,134)(109,143)(110,136)(111,129)(112,138)(162,170)(164,172)(166,174)(168,176)(177,232)(178,225)(179,234)(180,227)(181,236)(182,229)(183,238)(184,231)(185,240)(186,233)(187,226)(188,235)(189,228)(190,237)(191,230)(192,239)(210,218)(212,220)(214,222)(216,224)>;
G:=Group( (1,39,236,209,126,138,93,181,203,52,112,156,161,74,26)(2,204,40,53,237,97,210,157,127,162,139,75,94,27,182)(3,128,205,163,41,140,54,76,238,95,98,28,211,183,158)(4,239,113,96,206,99,164,29,42,212,141,184,55,159,77)(5,43,240,213,114,142,81,185,207,56,100,160,165,78,30)(6,208,44,57,225,101,214,145,115,166,143,79,82,31,186)(7,116,193,167,45,144,58,80,226,83,102,32,215,187,146)(8,227,117,84,194,103,168,17,46,216,129,188,59,147,65)(9,47,228,217,118,130,85,189,195,60,104,148,169,66,18)(10,196,48,61,229,105,218,149,119,170,131,67,86,19,190)(11,120,197,171,33,132,62,68,230,87,106,20,219,191,150)(12,231,121,88,198,107,172,21,34,220,133,192,63,151,69)(13,35,232,221,122,134,89,177,199,64,108,152,173,70,22)(14,200,36,49,233,109,222,153,123,174,135,71,90,23,178)(15,124,201,175,37,136,50,72,234,91,110,24,223,179,154)(16,235,125,92,202,111,176,25,38,224,137,180,51,155,73), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (2,10)(4,12)(6,14)(8,16)(17,125)(18,118)(19,127)(20,120)(21,113)(22,122)(23,115)(24,124)(25,117)(26,126)(27,119)(28,128)(29,121)(30,114)(31,123)(32,116)(33,150)(34,159)(35,152)(36,145)(37,154)(38,147)(39,156)(40,149)(41,158)(42,151)(43,160)(44,153)(45,146)(46,155)(47,148)(48,157)(49,57)(51,59)(53,61)(55,63)(65,202)(66,195)(67,204)(68,197)(69,206)(70,199)(71,208)(72,201)(73,194)(74,203)(75,196)(76,205)(77,198)(78,207)(79,200)(80,193)(82,90)(84,92)(86,94)(88,96)(97,131)(98,140)(99,133)(100,142)(101,135)(102,144)(103,137)(104,130)(105,139)(106,132)(107,141)(108,134)(109,143)(110,136)(111,129)(112,138)(162,170)(164,172)(166,174)(168,176)(177,232)(178,225)(179,234)(180,227)(181,236)(182,229)(183,238)(184,231)(185,240)(186,233)(187,226)(188,235)(189,228)(190,237)(191,230)(192,239)(210,218)(212,220)(214,222)(216,224) );
G=PermutationGroup([[(1,39,236,209,126,138,93,181,203,52,112,156,161,74,26),(2,204,40,53,237,97,210,157,127,162,139,75,94,27,182),(3,128,205,163,41,140,54,76,238,95,98,28,211,183,158),(4,239,113,96,206,99,164,29,42,212,141,184,55,159,77),(5,43,240,213,114,142,81,185,207,56,100,160,165,78,30),(6,208,44,57,225,101,214,145,115,166,143,79,82,31,186),(7,116,193,167,45,144,58,80,226,83,102,32,215,187,146),(8,227,117,84,194,103,168,17,46,216,129,188,59,147,65),(9,47,228,217,118,130,85,189,195,60,104,148,169,66,18),(10,196,48,61,229,105,218,149,119,170,131,67,86,19,190),(11,120,197,171,33,132,62,68,230,87,106,20,219,191,150),(12,231,121,88,198,107,172,21,34,220,133,192,63,151,69),(13,35,232,221,122,134,89,177,199,64,108,152,173,70,22),(14,200,36,49,233,109,222,153,123,174,135,71,90,23,178),(15,124,201,175,37,136,50,72,234,91,110,24,223,179,154),(16,235,125,92,202,111,176,25,38,224,137,180,51,155,73)], [(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),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(2,10),(4,12),(6,14),(8,16),(17,125),(18,118),(19,127),(20,120),(21,113),(22,122),(23,115),(24,124),(25,117),(26,126),(27,119),(28,128),(29,121),(30,114),(31,123),(32,116),(33,150),(34,159),(35,152),(36,145),(37,154),(38,147),(39,156),(40,149),(41,158),(42,151),(43,160),(44,153),(45,146),(46,155),(47,148),(48,157),(49,57),(51,59),(53,61),(55,63),(65,202),(66,195),(67,204),(68,197),(69,206),(70,199),(71,208),(72,201),(73,194),(74,203),(75,196),(76,205),(77,198),(78,207),(79,200),(80,193),(82,90),(84,92),(86,94),(88,96),(97,131),(98,140),(99,133),(100,142),(101,135),(102,144),(103,137),(104,130),(105,139),(106,132),(107,141),(108,134),(109,143),(110,136),(111,129),(112,138),(162,170),(164,172),(166,174),(168,176),(177,232),(178,225),(179,234),(180,227),(181,236),(182,229),(183,238),(184,231),(185,240),(186,233),(187,226),(188,235),(189,228),(190,237),(191,230),(192,239),(210,218),(212,220),(214,222),(216,224)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 12A | 12B | 15 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30 | 48A | ··· | 48H | 60A | 60B |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 48 | ··· | 48 | 60 | 60 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 4 | 2 | 5 | 5 | 5 | 5 | 30 | 30 | 4 | 12 | 12 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 12 | 12 | 10 | 10 | 10 | 10 | 8 | 10 | ··· | 10 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | C4×S3 | M5(2) | S3×C8 | D6.C8 | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C20.C8 | S3×F5 | S3×C5⋊C8 | C15⋊M5(2) |
| kernel | C15⋊M5(2) | C3×C5⋊C16 | C15⋊C16 | S3×C5⋊2C8 | C15⋊3C8 | S3×C20 | C5×Dic3 | S3×C10 | C5⋊C16 | C5⋊2C8 | C20 | C15 | C10 | C5 | C4×S3 | Dic3 | C12 | D6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 |
Matrix representation of C15⋊M5(2) ►in GL8(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 81 | 0 | 189 | 51 |
| 0 | 0 | 0 | 0 | 0 | 81 | 189 | 0 |
| 195 | 239 | 0 | 0 | 0 | 0 | 0 | 0 |
| 109 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 170 | 169 | 42 | 237 |
| 0 | 0 | 0 | 0 | 178 | 175 | 48 | 27 |
| 0 | 0 | 0 | 0 | 19 | 58 | 129 | 178 |
| 0 | 0 | 0 | 0 | 73 | 51 | 227 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 195 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,52,52,81,0,0,0,0,0,189,240,0,81,0,0,0,0,0,0,189,189,0,0,0,0,0,0,51,0],[195,109,0,0,0,0,0,0,239,46,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,170,178,19,73,0,0,0,0,169,175,58,51,0,0,0,0,42,48,129,227,0,0,0,0,237,27,178,8],[1,195,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C15⋊M5(2) in GAP, Magma, Sage, TeX
C_{15}\rtimes M_5(2) % in TeX
G:=Group("C15:M5(2)"); // GroupNames label
G:=SmallGroup(480,241);
// by ID
G=gap.SmallGroup(480,241);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,b*a*b^-1=a^8,c*a*c=a^11,c*b*c=b^9>;
// generators/relations
Export