metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊7D16, D8⋊1D15, C8.4D30, C60.3D4, D120⋊6C2, C40.12D6, C30.39D8, C24.12D10, C120.9C22, (C5×D8)⋊1S3, (C3×D8)⋊1D5, (C15×D8)⋊1C2, C3⋊3(C5⋊D16), C5⋊3(C3⋊D16), C15⋊3C16⋊1C2, C6.17(D4⋊D5), C2.4(D4⋊D15), C4.1(C15⋊7D4), C10.17(D4⋊S3), C12.17(C5⋊D4), C20.15(C3⋊D4), SmallGroup(480,186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊7D16
G = < a,b,c | a15=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
8C2
120C2
4C22
60C22
8C6
40S3
8C10
24D5
2D4
30D4
20D6
12D10
8C30
8D15
15D8
15C16
10D12
6D20
4D30
15D16
5D24
3D40
2D60
Smallest permutation representation of C15⋊7D16
►On 240 points
Generators in S240
(1 157 194 134 59 112 189 86 73 218 176 126 45 240 23)(2 24 225 46 127 161 219 74 87 190 97 60 135 195 158)(3 159 196 136 61 98 191 88 75 220 162 128 47 226 25)(4 26 227 48 113 163 221 76 89 192 99 62 137 197 160)(5 145 198 138 63 100 177 90 77 222 164 114 33 228 27)(6 28 229 34 115 165 223 78 91 178 101 64 139 199 146)(7 147 200 140 49 102 179 92 79 224 166 116 35 230 29)(8 30 231 36 117 167 209 80 93 180 103 50 141 201 148)(9 149 202 142 51 104 181 94 65 210 168 118 37 232 31)(10 32 233 38 119 169 211 66 95 182 105 52 143 203 150)(11 151 204 144 53 106 183 96 67 212 170 120 39 234 17)(12 18 235 40 121 171 213 68 81 184 107 54 129 205 152)(13 153 206 130 55 108 185 82 69 214 172 122 41 236 19)(14 20 237 42 123 173 215 70 83 186 109 56 131 207 154)(15 155 208 132 57 110 187 84 71 216 174 124 43 238 21)(16 22 239 44 125 175 217 72 85 188 111 58 133 193 156)
(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 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 147)(18 146)(19 145)(20 160)(21 159)(22 158)(23 157)(24 156)(25 155)(26 154)(27 153)(28 152)(29 151)(30 150)(31 149)(32 148)(33 130)(34 129)(35 144)(36 143)(37 142)(38 141)(39 140)(40 139)(41 138)(42 137)(43 136)(44 135)(45 134)(46 133)(47 132)(48 131)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 128)(58 127)(59 126)(60 125)(61 124)(62 123)(63 122)(64 121)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 96)(80 95)(97 175)(98 174)(99 173)(100 172)(101 171)(102 170)(103 169)(104 168)(105 167)(106 166)(107 165)(108 164)(109 163)(110 162)(111 161)(112 176)(177 214)(178 213)(179 212)(180 211)(181 210)(182 209)(183 224)(184 223)(185 222)(186 221)(187 220)(188 219)(189 218)(190 217)(191 216)(192 215)(193 225)(194 240)(195 239)(196 238)(197 237)(198 236)(199 235)(200 234)(201 233)(202 232)(203 231)(204 230)(205 229)(206 228)(207 227)(208 226)
G:=sub<Sym(240)| (1,157,194,134,59,112,189,86,73,218,176,126,45,240,23)(2,24,225,46,127,161,219,74,87,190,97,60,135,195,158)(3,159,196,136,61,98,191,88,75,220,162,128,47,226,25)(4,26,227,48,113,163,221,76,89,192,99,62,137,197,160)(5,145,198,138,63,100,177,90,77,222,164,114,33,228,27)(6,28,229,34,115,165,223,78,91,178,101,64,139,199,146)(7,147,200,140,49,102,179,92,79,224,166,116,35,230,29)(8,30,231,36,117,167,209,80,93,180,103,50,141,201,148)(9,149,202,142,51,104,181,94,65,210,168,118,37,232,31)(10,32,233,38,119,169,211,66,95,182,105,52,143,203,150)(11,151,204,144,53,106,183,96,67,212,170,120,39,234,17)(12,18,235,40,121,171,213,68,81,184,107,54,129,205,152)(13,153,206,130,55,108,185,82,69,214,172,122,41,236,19)(14,20,237,42,123,173,215,70,83,186,109,56,131,207,154)(15,155,208,132,57,110,187,84,71,216,174,124,43,238,21)(16,22,239,44,125,175,217,72,85,188,111,58,133,193,156), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,147)(18,146)(19,145)(20,160)(21,159)(22,158)(23,157)(24,156)(25,155)(26,154)(27,153)(28,152)(29,151)(30,150)(31,149)(32,148)(33,130)(34,129)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,96)(80,95)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,168)(105,167)(106,166)(107,165)(108,164)(109,163)(110,162)(111,161)(112,176)(177,214)(178,213)(179,212)(180,211)(181,210)(182,209)(183,224)(184,223)(185,222)(186,221)(187,220)(188,219)(189,218)(190,217)(191,216)(192,215)(193,225)(194,240)(195,239)(196,238)(197,237)(198,236)(199,235)(200,234)(201,233)(202,232)(203,231)(204,230)(205,229)(206,228)(207,227)(208,226)>;
G:=Group( (1,157,194,134,59,112,189,86,73,218,176,126,45,240,23)(2,24,225,46,127,161,219,74,87,190,97,60,135,195,158)(3,159,196,136,61,98,191,88,75,220,162,128,47,226,25)(4,26,227,48,113,163,221,76,89,192,99,62,137,197,160)(5,145,198,138,63,100,177,90,77,222,164,114,33,228,27)(6,28,229,34,115,165,223,78,91,178,101,64,139,199,146)(7,147,200,140,49,102,179,92,79,224,166,116,35,230,29)(8,30,231,36,117,167,209,80,93,180,103,50,141,201,148)(9,149,202,142,51,104,181,94,65,210,168,118,37,232,31)(10,32,233,38,119,169,211,66,95,182,105,52,143,203,150)(11,151,204,144,53,106,183,96,67,212,170,120,39,234,17)(12,18,235,40,121,171,213,68,81,184,107,54,129,205,152)(13,153,206,130,55,108,185,82,69,214,172,122,41,236,19)(14,20,237,42,123,173,215,70,83,186,109,56,131,207,154)(15,155,208,132,57,110,187,84,71,216,174,124,43,238,21)(16,22,239,44,125,175,217,72,85,188,111,58,133,193,156), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,147)(18,146)(19,145)(20,160)(21,159)(22,158)(23,157)(24,156)(25,155)(26,154)(27,153)(28,152)(29,151)(30,150)(31,149)(32,148)(33,130)(34,129)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,96)(80,95)(97,175)(98,174)(99,173)(100,172)(101,171)(102,170)(103,169)(104,168)(105,167)(106,166)(107,165)(108,164)(109,163)(110,162)(111,161)(112,176)(177,214)(178,213)(179,212)(180,211)(181,210)(182,209)(183,224)(184,223)(185,222)(186,221)(187,220)(188,219)(189,218)(190,217)(191,216)(192,215)(193,225)(194,240)(195,239)(196,238)(197,237)(198,236)(199,235)(200,234)(201,233)(202,232)(203,231)(204,230)(205,229)(206,228)(207,227)(208,226) );
G=PermutationGroup([[(1,157,194,134,59,112,189,86,73,218,176,126,45,240,23),(2,24,225,46,127,161,219,74,87,190,97,60,135,195,158),(3,159,196,136,61,98,191,88,75,220,162,128,47,226,25),(4,26,227,48,113,163,221,76,89,192,99,62,137,197,160),(5,145,198,138,63,100,177,90,77,222,164,114,33,228,27),(6,28,229,34,115,165,223,78,91,178,101,64,139,199,146),(7,147,200,140,49,102,179,92,79,224,166,116,35,230,29),(8,30,231,36,117,167,209,80,93,180,103,50,141,201,148),(9,149,202,142,51,104,181,94,65,210,168,118,37,232,31),(10,32,233,38,119,169,211,66,95,182,105,52,143,203,150),(11,151,204,144,53,106,183,96,67,212,170,120,39,234,17),(12,18,235,40,121,171,213,68,81,184,107,54,129,205,152),(13,153,206,130,55,108,185,82,69,214,172,122,41,236,19),(14,20,237,42,123,173,215,70,83,186,109,56,131,207,154),(15,155,208,132,57,110,187,84,71,216,174,124,43,238,21),(16,22,239,44,125,175,217,72,85,188,111,58,133,193,156)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,147),(18,146),(19,145),(20,160),(21,159),(22,158),(23,157),(24,156),(25,155),(26,154),(27,153),(28,152),(29,151),(30,150),(31,149),(32,148),(33,130),(34,129),(35,144),(36,143),(37,142),(38,141),(39,140),(40,139),(41,138),(42,137),(43,136),(44,135),(45,134),(46,133),(47,132),(48,131),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,128),(58,127),(59,126),(60,125),(61,124),(62,123),(63,122),(64,121),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,81),(79,96),(80,95),(97,175),(98,174),(99,173),(100,172),(101,171),(102,170),(103,169),(104,168),(105,167),(106,166),(107,165),(108,164),(109,163),(110,162),(111,161),(112,176),(177,214),(178,213),(179,212),(180,211),(181,210),(182,209),(183,224),(184,223),(185,222),(186,221),(187,220),(188,219),(189,218),(190,217),(191,216),(192,215),(193,225),(194,240),(195,239),(196,238),(197,237),(198,236),(199,235),(200,234),(201,233),(202,232),(203,231),(204,230),(205,229),(206,228),(207,227),(208,226)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 8 | 120 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | D15 | D16 | C5⋊D4 | D30 | C15⋊7D4 | D4⋊S3 | D4⋊D5 | C3⋊D16 | C5⋊D16 | D4⋊D15 | C15⋊7D16 |
| kernel | C15⋊7D16 | C15⋊3C16 | D120 | C15×D8 | C5×D8 | C60 | C3×D8 | C40 | C30 | C24 | C20 | D8 | C15 | C12 | C8 | C4 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊7D16 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 178 | 68 |
| 0 | 0 | 173 | 16 |
| 58 | 119 | 0 | 0 |
| 8 | 112 | 0 | 0 |
| 0 | 0 | 178 | 30 |
| 0 | 0 | 173 | 63 |
| 1 | 0 | 0 | 0 |
| 205 | 240 | 0 | 0 |
| 0 | 0 | 178 | 30 |
| 0 | 0 | 173 | 63 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,178,173,0,0,68,16],[58,8,0,0,119,112,0,0,0,0,178,173,0,0,30,63],[1,205,0,0,0,240,0,0,0,0,178,173,0,0,30,63] >;
C15⋊7D16 in GAP, Magma, Sage, TeX
C_{15}\rtimes_7D_{16} % in TeX
G:=Group("C15:7D16"); // GroupNames label
G:=SmallGroup(480,186);
// by ID
G=gap.SmallGroup(480,186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,254,135,142,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export