metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.93D4, C12.54D20, C30.14M4(2), (C6×D5)⋊1C8, D10⋊1(C3⋊C8), C6.14(C8×D5), C15⋊5(C22⋊C8), C30.28(C2×C8), C3⋊3(D10⋊1C8), (C2×C20).321D6, C6.8(C8⋊D5), (C6×Dic5).7C4, (C2×C12).325D10, C4.26(C3⋊D20), C5⋊3(C12.55D4), C12.85(C5⋊D4), C4.26(C15⋊D4), C20.84(C3⋊D4), C30.38(C22⋊C4), (C2×C60).223C22, (C2×Dic5).4Dic3, C10.7(C4.Dic3), C6.23(D10⋊C4), (C22×D5).3Dic3, C22.10(D5×Dic3), C2.1(D10⋊Dic3), C2.2(C20.32D6), C10.12(C6.D4), (C2×C3⋊C8)⋊9D5, (C10×C3⋊C8)⋊9C2, C2.4(D5×C3⋊C8), (D5×C2×C6).5C4, (C2×C4×D5).7S3, C10.13(C2×C3⋊C8), (D5×C2×C12).7C2, (C2×C6).44(C4×D5), (C2×C15⋊3C8)⋊21C2, (C2×C30).77(C2×C4), (C2×C4).226(S3×D5), (C2×C10).31(C2×Dic3), SmallGroup(480,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.93D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a29, cac=a49, cbc=a15b3 >
Subgroups: 380 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C3⋊C8, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C5⋊2C8, C2×C40, C2×C4×D5, C12.55D4, C5×C3⋊C8, C15⋊3C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D10⋊1C8, C10×C3⋊C8, C2×C15⋊3C8, D5×C2×C12, C60.93D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D5, Dic3, D6, C22⋊C4, C2×C8, M4(2), D10, C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C4×D5, D20, C5⋊D4, C2×C3⋊C8, C4.Dic3, C6.D4, S3×D5, C8×D5, C8⋊D5, D10⋊C4, C12.55D4, D5×Dic3, C15⋊D4, C3⋊D20, D10⋊1C8, D5×C3⋊C8, C20.32D6, D10⋊Dic3, C60.93D4
►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 99 163 202 31 69 133 232)(2 68 164 231 32 98 134 201)(3 97 165 200 33 67 135 230)(4 66 166 229 34 96 136 199)(5 95 167 198 35 65 137 228)(6 64 168 227 36 94 138 197)(7 93 169 196 37 63 139 226)(8 62 170 225 38 92 140 195)(9 91 171 194 39 61 141 224)(10 120 172 223 40 90 142 193)(11 89 173 192 41 119 143 222)(12 118 174 221 42 88 144 191)(13 87 175 190 43 117 145 220)(14 116 176 219 44 86 146 189)(15 85 177 188 45 115 147 218)(16 114 178 217 46 84 148 187)(17 83 179 186 47 113 149 216)(18 112 180 215 48 82 150 185)(19 81 121 184 49 111 151 214)(20 110 122 213 50 80 152 183)(21 79 123 182 51 109 153 212)(22 108 124 211 52 78 154 181)(23 77 125 240 53 107 155 210)(24 106 126 209 54 76 156 239)(25 75 127 238 55 105 157 208)(26 104 128 207 56 74 158 237)(27 73 129 236 57 103 159 206)(28 102 130 205 58 72 160 235)(29 71 131 234 59 101 161 204)(30 100 132 203 60 70 162 233)
(1 31)(2 20)(3 9)(4 58)(5 47)(6 36)(7 25)(8 14)(10 52)(11 41)(12 30)(13 19)(15 57)(16 46)(17 35)(18 24)(21 51)(22 40)(23 29)(26 56)(27 45)(28 34)(32 50)(33 39)(37 55)(38 44)(42 60)(43 49)(48 54)(53 59)(61 185)(62 234)(63 223)(64 212)(65 201)(66 190)(67 239)(68 228)(69 217)(70 206)(71 195)(72 184)(73 233)(74 222)(75 211)(76 200)(77 189)(78 238)(79 227)(80 216)(81 205)(82 194)(83 183)(84 232)(85 221)(86 210)(87 199)(88 188)(89 237)(90 226)(91 215)(92 204)(93 193)(94 182)(95 231)(96 220)(97 209)(98 198)(99 187)(100 236)(101 225)(102 214)(103 203)(104 192)(105 181)(106 230)(107 219)(108 208)(109 197)(110 186)(111 235)(112 224)(113 213)(114 202)(115 191)(116 240)(117 229)(118 218)(119 207)(120 196)(121 175)(122 164)(123 153)(124 142)(125 131)(126 180)(127 169)(128 158)(129 147)(130 136)(132 174)(133 163)(134 152)(135 141)(137 179)(138 168)(139 157)(140 146)(143 173)(144 162)(145 151)(148 178)(149 167)(150 156)(154 172)(155 161)(159 177)(160 166)(165 171)(170 176)
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), (1,99,163,202,31,69,133,232)(2,68,164,231,32,98,134,201)(3,97,165,200,33,67,135,230)(4,66,166,229,34,96,136,199)(5,95,167,198,35,65,137,228)(6,64,168,227,36,94,138,197)(7,93,169,196,37,63,139,226)(8,62,170,225,38,92,140,195)(9,91,171,194,39,61,141,224)(10,120,172,223,40,90,142,193)(11,89,173,192,41,119,143,222)(12,118,174,221,42,88,144,191)(13,87,175,190,43,117,145,220)(14,116,176,219,44,86,146,189)(15,85,177,188,45,115,147,218)(16,114,178,217,46,84,148,187)(17,83,179,186,47,113,149,216)(18,112,180,215,48,82,150,185)(19,81,121,184,49,111,151,214)(20,110,122,213,50,80,152,183)(21,79,123,182,51,109,153,212)(22,108,124,211,52,78,154,181)(23,77,125,240,53,107,155,210)(24,106,126,209,54,76,156,239)(25,75,127,238,55,105,157,208)(26,104,128,207,56,74,158,237)(27,73,129,236,57,103,159,206)(28,102,130,205,58,72,160,235)(29,71,131,234,59,101,161,204)(30,100,132,203,60,70,162,233), (1,31)(2,20)(3,9)(4,58)(5,47)(6,36)(7,25)(8,14)(10,52)(11,41)(12,30)(13,19)(15,57)(16,46)(17,35)(18,24)(21,51)(22,40)(23,29)(26,56)(27,45)(28,34)(32,50)(33,39)(37,55)(38,44)(42,60)(43,49)(48,54)(53,59)(61,185)(62,234)(63,223)(64,212)(65,201)(66,190)(67,239)(68,228)(69,217)(70,206)(71,195)(72,184)(73,233)(74,222)(75,211)(76,200)(77,189)(78,238)(79,227)(80,216)(81,205)(82,194)(83,183)(84,232)(85,221)(86,210)(87,199)(88,188)(89,237)(90,226)(91,215)(92,204)(93,193)(94,182)(95,231)(96,220)(97,209)(98,198)(99,187)(100,236)(101,225)(102,214)(103,203)(104,192)(105,181)(106,230)(107,219)(108,208)(109,197)(110,186)(111,235)(112,224)(113,213)(114,202)(115,191)(116,240)(117,229)(118,218)(119,207)(120,196)(121,175)(122,164)(123,153)(124,142)(125,131)(126,180)(127,169)(128,158)(129,147)(130,136)(132,174)(133,163)(134,152)(135,141)(137,179)(138,168)(139,157)(140,146)(143,173)(144,162)(145,151)(148,178)(149,167)(150,156)(154,172)(155,161)(159,177)(160,166)(165,171)(170,176)>;
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), (1,99,163,202,31,69,133,232)(2,68,164,231,32,98,134,201)(3,97,165,200,33,67,135,230)(4,66,166,229,34,96,136,199)(5,95,167,198,35,65,137,228)(6,64,168,227,36,94,138,197)(7,93,169,196,37,63,139,226)(8,62,170,225,38,92,140,195)(9,91,171,194,39,61,141,224)(10,120,172,223,40,90,142,193)(11,89,173,192,41,119,143,222)(12,118,174,221,42,88,144,191)(13,87,175,190,43,117,145,220)(14,116,176,219,44,86,146,189)(15,85,177,188,45,115,147,218)(16,114,178,217,46,84,148,187)(17,83,179,186,47,113,149,216)(18,112,180,215,48,82,150,185)(19,81,121,184,49,111,151,214)(20,110,122,213,50,80,152,183)(21,79,123,182,51,109,153,212)(22,108,124,211,52,78,154,181)(23,77,125,240,53,107,155,210)(24,106,126,209,54,76,156,239)(25,75,127,238,55,105,157,208)(26,104,128,207,56,74,158,237)(27,73,129,236,57,103,159,206)(28,102,130,205,58,72,160,235)(29,71,131,234,59,101,161,204)(30,100,132,203,60,70,162,233), (1,31)(2,20)(3,9)(4,58)(5,47)(6,36)(7,25)(8,14)(10,52)(11,41)(12,30)(13,19)(15,57)(16,46)(17,35)(18,24)(21,51)(22,40)(23,29)(26,56)(27,45)(28,34)(32,50)(33,39)(37,55)(38,44)(42,60)(43,49)(48,54)(53,59)(61,185)(62,234)(63,223)(64,212)(65,201)(66,190)(67,239)(68,228)(69,217)(70,206)(71,195)(72,184)(73,233)(74,222)(75,211)(76,200)(77,189)(78,238)(79,227)(80,216)(81,205)(82,194)(83,183)(84,232)(85,221)(86,210)(87,199)(88,188)(89,237)(90,226)(91,215)(92,204)(93,193)(94,182)(95,231)(96,220)(97,209)(98,198)(99,187)(100,236)(101,225)(102,214)(103,203)(104,192)(105,181)(106,230)(107,219)(108,208)(109,197)(110,186)(111,235)(112,224)(113,213)(114,202)(115,191)(116,240)(117,229)(118,218)(119,207)(120,196)(121,175)(122,164)(123,153)(124,142)(125,131)(126,180)(127,169)(128,158)(129,147)(130,136)(132,174)(133,163)(134,152)(135,141)(137,179)(138,168)(139,157)(140,146)(143,173)(144,162)(145,151)(148,178)(149,167)(150,156)(154,172)(155,161)(159,177)(160,166)(165,171)(170,176) );
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)], [(1,99,163,202,31,69,133,232),(2,68,164,231,32,98,134,201),(3,97,165,200,33,67,135,230),(4,66,166,229,34,96,136,199),(5,95,167,198,35,65,137,228),(6,64,168,227,36,94,138,197),(7,93,169,196,37,63,139,226),(8,62,170,225,38,92,140,195),(9,91,171,194,39,61,141,224),(10,120,172,223,40,90,142,193),(11,89,173,192,41,119,143,222),(12,118,174,221,42,88,144,191),(13,87,175,190,43,117,145,220),(14,116,176,219,44,86,146,189),(15,85,177,188,45,115,147,218),(16,114,178,217,46,84,148,187),(17,83,179,186,47,113,149,216),(18,112,180,215,48,82,150,185),(19,81,121,184,49,111,151,214),(20,110,122,213,50,80,152,183),(21,79,123,182,51,109,153,212),(22,108,124,211,52,78,154,181),(23,77,125,240,53,107,155,210),(24,106,126,209,54,76,156,239),(25,75,127,238,55,105,157,208),(26,104,128,207,56,74,158,237),(27,73,129,236,57,103,159,206),(28,102,130,205,58,72,160,235),(29,71,131,234,59,101,161,204),(30,100,132,203,60,70,162,233)], [(1,31),(2,20),(3,9),(4,58),(5,47),(6,36),(7,25),(8,14),(10,52),(11,41),(12,30),(13,19),(15,57),(16,46),(17,35),(18,24),(21,51),(22,40),(23,29),(26,56),(27,45),(28,34),(32,50),(33,39),(37,55),(38,44),(42,60),(43,49),(48,54),(53,59),(61,185),(62,234),(63,223),(64,212),(65,201),(66,190),(67,239),(68,228),(69,217),(70,206),(71,195),(72,184),(73,233),(74,222),(75,211),(76,200),(77,189),(78,238),(79,227),(80,216),(81,205),(82,194),(83,183),(84,232),(85,221),(86,210),(87,199),(88,188),(89,237),(90,226),(91,215),(92,204),(93,193),(94,182),(95,231),(96,220),(97,209),(98,198),(99,187),(100,236),(101,225),(102,214),(103,203),(104,192),(105,181),(106,230),(107,219),(108,208),(109,197),(110,186),(111,235),(112,224),(113,213),(114,202),(115,191),(116,240),(117,229),(118,218),(119,207),(120,196),(121,175),(122,164),(123,153),(124,142),(125,131),(126,180),(127,169),(128,158),(129,147),(130,136),(132,174),(133,163),(134,152),(135,141),(137,179),(138,168),(139,157),(140,146),(143,173),(144,162),(145,151),(148,178),(149,167),(150,156),(154,172),(155,161),(159,177),(160,166),(165,171),(170,176)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | C8 | S3 | D4 | D5 | Dic3 | D6 | Dic3 | M4(2) | D10 | C3⋊D4 | C3⋊C8 | D20 | C5⋊D4 | C4×D5 | C4.Dic3 | C8×D5 | C8⋊D5 | S3×D5 | C15⋊D4 | C3⋊D20 | D5×Dic3 | D5×C3⋊C8 | C20.32D6 |
| kernel | C60.93D4 | C10×C3⋊C8 | C2×C15⋊3C8 | D5×C2×C12 | C6×Dic5 | D5×C2×C6 | C6×D5 | C2×C4×D5 | C60 | C2×C3⋊C8 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×C12 | C20 | D10 | C12 | C12 | C2×C6 | C10 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.93D4 ►in GL4(𝔽241) generated by
| 189 | 189 | 0 | 0 |
| 52 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 81 | 60 |
| 200 | 156 | 0 | 0 |
| 119 | 41 | 0 | 0 |
| 0 | 0 | 214 | 142 |
| 0 | 0 | 64 | 27 |
| 1 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [189,52,0,0,189,1,0,0,0,0,4,81,0,0,0,60],[200,119,0,0,156,41,0,0,0,0,214,64,0,0,142,27],[1,189,0,0,0,240,0,0,0,0,240,0,0,0,0,240] >;
C60.93D4 in GAP, Magma, Sage, TeX
C_{60}._{93}D_4 % in TeX
G:=Group("C60.93D4"); // GroupNames label
G:=SmallGroup(480,31);
// by ID
G=gap.SmallGroup(480,31);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^29,c*a*c=a^49,c*b*c=a^15*b^3>;
// generators/relations