metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60.5C4, Dic30.5C4, C60.183C23, C5⋊6(C8○D12), C3⋊C8.26D10, C15⋊15(C8○D4), C20.53(C4×S3), C12.34(C4×D5), (C2×C20).81D6, C15⋊7D4.4C4, C5⋊2C8.39D6, C60.116(C2×C4), D30.20(C2×C4), C4.Dic3⋊11D5, D15⋊2C8⋊11C2, (C2×C12).317D10, C3⋊3(D20.2C4), (C2×C60).48C22, C4.9(D30.C2), D30.5C4⋊12C2, D60⋊11C2.7C2, C20.180(C22×S3), C30.105(C22×C4), Dic15.21(C2×C4), (C4×D15).40C22, C12.180(C22×D5), C22.2(D30.C2), (C6×C5⋊2C8)⋊2C2, C6.39(C2×C4×D5), (C2×C5⋊2C8)⋊4S3, C10.72(S3×C2×C4), C4.153(C2×S3×D5), (C2×C6).21(C4×D5), (C2×C10).32(C4×S3), (C2×C4).144(S3×D5), (C5×C3⋊C8).26C22, (C5×C4.Dic3)⋊6C2, C2.4(C2×D30.C2), (C2×C30).102(C2×C4), (C3×C5⋊2C8).44C22, SmallGroup(480,366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60.5C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a49, cbc-1=a48b >
Subgroups: 572 in 124 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, D15, C30, C30, C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, Dic15, C60, D30, C2×C30, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, C8○D12, C5×C3⋊C8, C3×C5⋊2C8, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20.2C4, D15⋊2C8, D30.5C4, C6×C5⋊2C8, C5×C4.Dic3, D60⋊11C2, D60.5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C8○D4, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C8○D12, D30.C2, C2×S3×D5, D20.2C4, C2×D30.C2, D60.5C4
►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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(110 120)(111 119)(112 118)(113 117)(114 116)(121 129)(122 128)(123 127)(124 126)(130 180)(131 179)(132 178)(133 177)(134 176)(135 175)(136 174)(137 173)(138 172)(139 171)(140 170)(141 169)(142 168)(143 167)(144 166)(145 165)(146 164)(147 163)(148 162)(149 161)(150 160)(151 159)(152 158)(153 157)(154 156)(181 201)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)(190 192)(202 240)(203 239)(204 238)(205 237)(206 236)(207 235)(208 234)(209 233)(210 232)(211 231)(212 230)(213 229)(214 228)(215 227)(216 226)(217 225)(218 224)(219 223)(220 222)
(1 229 63 133 31 199 93 163)(2 218 64 122 32 188 94 152)(3 207 65 171 33 237 95 141)(4 196 66 160 34 226 96 130)(5 185 67 149 35 215 97 179)(6 234 68 138 36 204 98 168)(7 223 69 127 37 193 99 157)(8 212 70 176 38 182 100 146)(9 201 71 165 39 231 101 135)(10 190 72 154 40 220 102 124)(11 239 73 143 41 209 103 173)(12 228 74 132 42 198 104 162)(13 217 75 121 43 187 105 151)(14 206 76 170 44 236 106 140)(15 195 77 159 45 225 107 129)(16 184 78 148 46 214 108 178)(17 233 79 137 47 203 109 167)(18 222 80 126 48 192 110 156)(19 211 81 175 49 181 111 145)(20 200 82 164 50 230 112 134)(21 189 83 153 51 219 113 123)(22 238 84 142 52 208 114 172)(23 227 85 131 53 197 115 161)(24 216 86 180 54 186 116 150)(25 205 87 169 55 235 117 139)(26 194 88 158 56 224 118 128)(27 183 89 147 57 213 119 177)(28 232 90 136 58 202 120 166)(29 221 91 125 59 191 61 155)(30 210 92 174 60 240 62 144)
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(110,120)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,166)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222), (1,229,63,133,31,199,93,163)(2,218,64,122,32,188,94,152)(3,207,65,171,33,237,95,141)(4,196,66,160,34,226,96,130)(5,185,67,149,35,215,97,179)(6,234,68,138,36,204,98,168)(7,223,69,127,37,193,99,157)(8,212,70,176,38,182,100,146)(9,201,71,165,39,231,101,135)(10,190,72,154,40,220,102,124)(11,239,73,143,41,209,103,173)(12,228,74,132,42,198,104,162)(13,217,75,121,43,187,105,151)(14,206,76,170,44,236,106,140)(15,195,77,159,45,225,107,129)(16,184,78,148,46,214,108,178)(17,233,79,137,47,203,109,167)(18,222,80,126,48,192,110,156)(19,211,81,175,49,181,111,145)(20,200,82,164,50,230,112,134)(21,189,83,153,51,219,113,123)(22,238,84,142,52,208,114,172)(23,227,85,131,53,197,115,161)(24,216,86,180,54,186,116,150)(25,205,87,169,55,235,117,139)(26,194,88,158,56,224,118,128)(27,183,89,147,57,213,119,177)(28,232,90,136,58,202,120,166)(29,221,91,125,59,191,61,155)(30,210,92,174,60,240,62,144)>;
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)(110,120)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,166)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222), (1,229,63,133,31,199,93,163)(2,218,64,122,32,188,94,152)(3,207,65,171,33,237,95,141)(4,196,66,160,34,226,96,130)(5,185,67,149,35,215,97,179)(6,234,68,138,36,204,98,168)(7,223,69,127,37,193,99,157)(8,212,70,176,38,182,100,146)(9,201,71,165,39,231,101,135)(10,190,72,154,40,220,102,124)(11,239,73,143,41,209,103,173)(12,228,74,132,42,198,104,162)(13,217,75,121,43,187,105,151)(14,206,76,170,44,236,106,140)(15,195,77,159,45,225,107,129)(16,184,78,148,46,214,108,178)(17,233,79,137,47,203,109,167)(18,222,80,126,48,192,110,156)(19,211,81,175,49,181,111,145)(20,200,82,164,50,230,112,134)(21,189,83,153,51,219,113,123)(22,238,84,142,52,208,114,172)(23,227,85,131,53,197,115,161)(24,216,86,180,54,186,116,150)(25,205,87,169,55,235,117,139)(26,194,88,158,56,224,118,128)(27,183,89,147,57,213,119,177)(28,232,90,136,58,202,120,166)(29,221,91,125,59,191,61,155)(30,210,92,174,60,240,62,144) );
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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86),(110,120),(111,119),(112,118),(113,117),(114,116),(121,129),(122,128),(123,127),(124,126),(130,180),(131,179),(132,178),(133,177),(134,176),(135,175),(136,174),(137,173),(138,172),(139,171),(140,170),(141,169),(142,168),(143,167),(144,166),(145,165),(146,164),(147,163),(148,162),(149,161),(150,160),(151,159),(152,158),(153,157),(154,156),(181,201),(182,200),(183,199),(184,198),(185,197),(186,196),(187,195),(188,194),(189,193),(190,192),(202,240),(203,239),(204,238),(205,237),(206,236),(207,235),(208,234),(209,233),(210,232),(211,231),(212,230),(213,229),(214,228),(215,227),(216,226),(217,225),(218,224),(219,223),(220,222)], [(1,229,63,133,31,199,93,163),(2,218,64,122,32,188,94,152),(3,207,65,171,33,237,95,141),(4,196,66,160,34,226,96,130),(5,185,67,149,35,215,97,179),(6,234,68,138,36,204,98,168),(7,223,69,127,37,193,99,157),(8,212,70,176,38,182,100,146),(9,201,71,165,39,231,101,135),(10,190,72,154,40,220,102,124),(11,239,73,143,41,209,103,173),(12,228,74,132,42,198,104,162),(13,217,75,121,43,187,105,151),(14,206,76,170,44,236,106,140),(15,195,77,159,45,225,107,129),(16,184,78,148,46,214,108,178),(17,233,79,137,47,203,109,167),(18,222,80,126,48,192,110,156),(19,211,81,175,49,181,111,145),(20,200,82,164,50,230,112,134),(21,189,83,153,51,219,113,123),(22,238,84,142,52,208,114,172),(23,227,85,131,53,197,115,161),(24,216,86,180,54,186,116,150),(25,205,87,169,55,235,117,139),(26,194,88,158,56,224,118,128),(27,183,89,147,57,213,119,177),(28,232,90,136,58,202,120,166),(29,221,91,125,59,191,61,155),(30,210,92,174,60,240,62,144)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | ··· | 24H | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 30 | 30 | 2 | 1 | 1 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×S3 | C8○D4 | C4×D5 | C4×D5 | C8○D12 | S3×D5 | D30.C2 | C2×S3×D5 | D30.C2 | D20.2C4 | D60.5C4 |
| kernel | D60.5C4 | D15⋊2C8 | D30.5C4 | C6×C5⋊2C8 | C5×C4.Dic3 | D60⋊11C2 | Dic30 | D60 | C15⋊7D4 | C2×C5⋊2C8 | C4.Dic3 | C5⋊2C8 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C15 | C12 | C2×C6 | C5 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D60.5C4 ►in GL4(𝔽241) generated by
| 0 | 52 | 0 | 0 |
| 190 | 190 | 0 | 0 |
| 0 | 0 | 198 | 142 |
| 0 | 0 | 99 | 99 |
| 1 | 0 | 0 | 0 |
| 50 | 240 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 240 |
| 51 | 52 | 0 | 0 |
| 191 | 190 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(241))| [0,190,0,0,52,190,0,0,0,0,198,99,0,0,142,99],[1,50,0,0,0,240,0,0,0,0,1,0,0,0,240,240],[51,191,0,0,52,190,0,0,0,0,30,0,0,0,0,30] >;
D60.5C4 in GAP, Magma, Sage, TeX
D_{60}._5C_4 % in TeX
G:=Group("D60.5C4"); // GroupNames label
G:=SmallGroup(480,366);
// by ID
G=gap.SmallGroup(480,366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,219,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^48*b>;
// generators/relations