metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊9C4, C60.1D4, C4.9D60, C30.38D8, C12.9D20, C20.9D12, C30.33SD16, C4⋊C4⋊1D15, C4.1(C4×D15), C20.47(C4×S3), C60.77(C2×C4), (C2×D60).6C2, C12.15(C4×D5), (C2×C4).34D30, (C2×C20).66D6, C5⋊3(C6.D8), C6.8(Q8⋊D5), C3⋊2(D20⋊6C4), C2.2(D4⋊D15), C6.16(D4⋊D5), (C2×C30).137D4, (C2×C12).67D10, C15⋊11(D4⋊C4), C10.16(D4⋊S3), C10.29(D6⋊C4), (C2×C60).52C22, C2.5(D30⋊3C4), C30.71(C22⋊C4), C10.8(Q8⋊2S3), C2.2(Q8⋊2D15), C6.14(D10⋊C4), C22.14(C15⋊7D4), (C5×C4⋊C4)⋊1S3, (C3×C4⋊C4)⋊1D5, (C15×C4⋊C4)⋊1C2, (C2×C15⋊3C8)⋊1C2, (C2×C6).69(C5⋊D4), (C2×C10).69(C3⋊D4), SmallGroup(480,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊9C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a15b >
Subgroups: 804 in 100 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, D10, C2×C10, C3⋊C8, D12, C2×C12, C2×C12, C22×S3, D15, C30, D4⋊C4, C5⋊2C8, D20, C2×C20, C2×C20, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C60, C60, D30, C2×C30, C2×C5⋊2C8, C5×C4⋊C4, C2×D20, C6.D8, C15⋊3C8, D60, D60, C2×C60, C2×C60, C22×D15, D20⋊6C4, C2×C15⋊3C8, C15×C4⋊C4, C2×D60, D60⋊9C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D15, D4⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D4⋊S3, Q8⋊2S3, D30, D10⋊C4, D4⋊D5, Q8⋊D5, C6.D8, C4×D15, D60, C15⋊7D4, D20⋊6C4, D30⋊3C4, D4⋊D15, Q8⋊2D15, D60⋊9C4
►On 240 points
Generators in S240
(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)
(1 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(140 180)(141 179)(142 178)(143 177)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 240)(197 239)(198 238)(199 237)(200 236)(201 235)(202 234)(203 233)(204 232)(205 231)(206 230)(207 229)(208 228)(209 227)(210 226)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 226 113 168)(2 197 114 139)(3 228 115 170)(4 199 116 141)(5 230 117 172)(6 201 118 143)(7 232 119 174)(8 203 120 145)(9 234 61 176)(10 205 62 147)(11 236 63 178)(12 207 64 149)(13 238 65 180)(14 209 66 151)(15 240 67 122)(16 211 68 153)(17 182 69 124)(18 213 70 155)(19 184 71 126)(20 215 72 157)(21 186 73 128)(22 217 74 159)(23 188 75 130)(24 219 76 161)(25 190 77 132)(26 221 78 163)(27 192 79 134)(28 223 80 165)(29 194 81 136)(30 225 82 167)(31 196 83 138)(32 227 84 169)(33 198 85 140)(34 229 86 171)(35 200 87 142)(36 231 88 173)(37 202 89 144)(38 233 90 175)(39 204 91 146)(40 235 92 177)(41 206 93 148)(42 237 94 179)(43 208 95 150)(44 239 96 121)(45 210 97 152)(46 181 98 123)(47 212 99 154)(48 183 100 125)(49 214 101 156)(50 185 102 127)(51 216 103 158)(52 187 104 129)(53 218 105 160)(54 189 106 131)(55 220 107 162)(56 191 108 133)(57 222 109 164)(58 193 110 135)(59 224 111 166)(60 195 112 137)
G:=sub<Sym(240)| (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,240)(197,239)(198,238)(199,237)(200,236)(201,235)(202,234)(203,233)(204,232)(205,231)(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,226,113,168)(2,197,114,139)(3,228,115,170)(4,199,116,141)(5,230,117,172)(6,201,118,143)(7,232,119,174)(8,203,120,145)(9,234,61,176)(10,205,62,147)(11,236,63,178)(12,207,64,149)(13,238,65,180)(14,209,66,151)(15,240,67,122)(16,211,68,153)(17,182,69,124)(18,213,70,155)(19,184,71,126)(20,215,72,157)(21,186,73,128)(22,217,74,159)(23,188,75,130)(24,219,76,161)(25,190,77,132)(26,221,78,163)(27,192,79,134)(28,223,80,165)(29,194,81,136)(30,225,82,167)(31,196,83,138)(32,227,84,169)(33,198,85,140)(34,229,86,171)(35,200,87,142)(36,231,88,173)(37,202,89,144)(38,233,90,175)(39,204,91,146)(40,235,92,177)(41,206,93,148)(42,237,94,179)(43,208,95,150)(44,239,96,121)(45,210,97,152)(46,181,98,123)(47,212,99,154)(48,183,100,125)(49,214,101,156)(50,185,102,127)(51,216,103,158)(52,187,104,129)(53,218,105,160)(54,189,106,131)(55,220,107,162)(56,191,108,133)(57,222,109,164)(58,193,110,135)(59,224,111,166)(60,195,112,137)>;
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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,240)(197,239)(198,238)(199,237)(200,236)(201,235)(202,234)(203,233)(204,232)(205,231)(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,226,113,168)(2,197,114,139)(3,228,115,170)(4,199,116,141)(5,230,117,172)(6,201,118,143)(7,232,119,174)(8,203,120,145)(9,234,61,176)(10,205,62,147)(11,236,63,178)(12,207,64,149)(13,238,65,180)(14,209,66,151)(15,240,67,122)(16,211,68,153)(17,182,69,124)(18,213,70,155)(19,184,71,126)(20,215,72,157)(21,186,73,128)(22,217,74,159)(23,188,75,130)(24,219,76,161)(25,190,77,132)(26,221,78,163)(27,192,79,134)(28,223,80,165)(29,194,81,136)(30,225,82,167)(31,196,83,138)(32,227,84,169)(33,198,85,140)(34,229,86,171)(35,200,87,142)(36,231,88,173)(37,202,89,144)(38,233,90,175)(39,204,91,146)(40,235,92,177)(41,206,93,148)(42,237,94,179)(43,208,95,150)(44,239,96,121)(45,210,97,152)(46,181,98,123)(47,212,99,154)(48,183,100,125)(49,214,101,156)(50,185,102,127)(51,216,103,158)(52,187,104,129)(53,218,105,160)(54,189,106,131)(55,220,107,162)(56,191,108,133)(57,222,109,164)(58,193,110,135)(59,224,111,166)(60,195,112,137) );
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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(121,139),(122,138),(123,137),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(140,180),(141,179),(142,178),(143,177),(144,176),(145,175),(146,174),(147,173),(148,172),(149,171),(150,170),(151,169),(152,168),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189),(196,240),(197,239),(198,238),(199,237),(200,236),(201,235),(202,234),(203,233),(204,232),(205,231),(206,230),(207,229),(208,228),(209,227),(210,226),(211,225),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)], [(1,226,113,168),(2,197,114,139),(3,228,115,170),(4,199,116,141),(5,230,117,172),(6,201,118,143),(7,232,119,174),(8,203,120,145),(9,234,61,176),(10,205,62,147),(11,236,63,178),(12,207,64,149),(13,238,65,180),(14,209,66,151),(15,240,67,122),(16,211,68,153),(17,182,69,124),(18,213,70,155),(19,184,71,126),(20,215,72,157),(21,186,73,128),(22,217,74,159),(23,188,75,130),(24,219,76,161),(25,190,77,132),(26,221,78,163),(27,192,79,134),(28,223,80,165),(29,194,81,136),(30,225,82,167),(31,196,83,138),(32,227,84,169),(33,198,85,140),(34,229,86,171),(35,200,87,142),(36,231,88,173),(37,202,89,144),(38,233,90,175),(39,204,91,146),(40,235,92,177),(41,206,93,148),(42,237,94,179),(43,208,95,150),(44,239,96,121),(45,210,97,152),(46,181,98,123),(47,212,99,154),(48,183,100,125),(49,214,101,156),(50,185,102,127),(51,216,103,158),(52,187,104,129),(53,218,105,160),(54,189,106,131),(55,220,107,162),(56,191,108,133),(57,222,109,164),(58,193,110,135),(59,224,111,166),(60,195,112,137)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | D10 | C4×S3 | D12 | C3⋊D4 | D15 | C4×D5 | D20 | C5⋊D4 | D30 | C4×D15 | D60 | C15⋊7D4 | D4⋊S3 | Q8⋊2S3 | D4⋊D5 | Q8⋊D5 | D4⋊D15 | Q8⋊2D15 |
| kernel | D60⋊9C4 | C2×C15⋊3C8 | C15×C4⋊C4 | C2×D60 | D60 | C5×C4⋊C4 | C60 | C2×C30 | C3×C4⋊C4 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D60⋊9C4 ►in GL6(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 93 | 30 | 0 | 0 |
| 0 | 0 | 17 | 148 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 171 | 140 | 0 | 0 | 0 | 0 |
| 101 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 197 | 156 | 0 | 0 |
| 0 | 0 | 88 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 219 |
| 0 | 0 | 0 | 0 | 230 | 0 |
G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,1,0,0,0,0,0,0,0,190,0,0,0,0,52,190,0,0,0,0,0,0,1,240,0,0,0,0,2,240],[240,0,0,0,0,0,1,1,0,0,0,0,0,0,93,17,0,0,0,0,30,148,0,0,0,0,0,0,1,240,0,0,0,0,0,240],[171,101,0,0,0,0,140,70,0,0,0,0,0,0,197,88,0,0,0,0,156,44,0,0,0,0,0,0,0,230,0,0,0,0,219,0] >;
D60⋊9C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_9C_4 % in TeX
G:=Group("D60:9C4"); // GroupNames label
G:=SmallGroup(480,169);
// by ID
G=gap.SmallGroup(480,169);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^15*b>;
// generators/relations