metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60.3C4, C40.36D6, C8.12D30, C24.39D10, M4(2)⋊5D15, Dic30.3C4, C60.256C23, C120.61C22, C4.5(C4×D15), (C8×D15)⋊17C2, C15⋊24(C8○D4), C20.62(C4×S3), C60.87(C2×C4), C5⋊7(D12.C4), (C2×C4).46D30, C12.30(C4×D5), C15⋊7D4.3C4, C40⋊S3⋊14C2, D30.25(C2×C4), (C2×C20).141D6, (C5×M4(2))⋊4S3, (C3×M4(2))⋊6D5, C22.1(C4×D15), (C2×C12).140D10, C3⋊4(D20.2C4), (C15×M4(2))⋊6C2, (C2×C60).67C22, C4.38(C22×D15), D60⋊11C2.9C2, C20.226(C22×S3), C30.167(C22×C4), C15⋊3C8.38C22, Dic15.32(C2×C4), (C4×D15).52C22, C12.228(C22×D5), C6.72(C2×C4×D5), C2.17(C2×C4×D15), (C2×C15⋊3C8)⋊3C2, C10.104(S3×C2×C4), (C2×C6).14(C4×D5), (C2×C10).37(C4×S3), (C2×C30).69(C2×C4), SmallGroup(480,872)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60.3C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a31, cbc-1=a30b >
Subgroups: 596 in 124 conjugacy classes, 55 normal (35 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), 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, C2×C3⋊C8, C3×M4(2), C4○D12, Dic15, C60, D30, C2×C30, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, D12.C4, C15⋊3C8, C120, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20.2C4, C8×D15, C40⋊S3, C2×C15⋊3C8, C15×M4(2), D60⋊11C2, D60.3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, D15, C8○D4, C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, D12.C4, C4×D15, C22×D15, D20.2C4, C2×C4×D15, D60.3C4
►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 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(121 172)(122 171)(123 170)(124 169)(125 168)(126 167)(127 166)(128 165)(129 164)(130 163)(131 162)(132 161)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(173 180)(174 179)(175 178)(176 177)(181 228)(182 227)(183 226)(184 225)(185 224)(186 223)(187 222)(188 221)(189 220)(190 219)(191 218)(192 217)(193 216)(194 215)(195 214)(196 213)(197 212)(198 211)(199 210)(200 209)(201 208)(202 207)(203 206)(204 205)(229 240)(230 239)(231 238)(232 237)(233 236)(234 235)
(1 67 235 177 31 97 205 147)(2 98 236 148 32 68 206 178)(3 69 237 179 33 99 207 149)(4 100 238 150 34 70 208 180)(5 71 239 121 35 101 209 151)(6 102 240 152 36 72 210 122)(7 73 181 123 37 103 211 153)(8 104 182 154 38 74 212 124)(9 75 183 125 39 105 213 155)(10 106 184 156 40 76 214 126)(11 77 185 127 41 107 215 157)(12 108 186 158 42 78 216 128)(13 79 187 129 43 109 217 159)(14 110 188 160 44 80 218 130)(15 81 189 131 45 111 219 161)(16 112 190 162 46 82 220 132)(17 83 191 133 47 113 221 163)(18 114 192 164 48 84 222 134)(19 85 193 135 49 115 223 165)(20 116 194 166 50 86 224 136)(21 87 195 137 51 117 225 167)(22 118 196 168 52 88 226 138)(23 89 197 139 53 119 227 169)(24 120 198 170 54 90 228 140)(25 91 199 141 55 61 229 171)(26 62 200 172 56 92 230 142)(27 93 201 143 57 63 231 173)(28 64 202 174 58 94 232 144)(29 95 203 145 59 65 233 175)(30 66 204 176 60 96 234 146)
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)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(173,180)(174,179)(175,178)(176,177)(181,228)(182,227)(183,226)(184,225)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)(200,209)(201,208)(202,207)(203,206)(204,205)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,67,235,177,31,97,205,147)(2,98,236,148,32,68,206,178)(3,69,237,179,33,99,207,149)(4,100,238,150,34,70,208,180)(5,71,239,121,35,101,209,151)(6,102,240,152,36,72,210,122)(7,73,181,123,37,103,211,153)(8,104,182,154,38,74,212,124)(9,75,183,125,39,105,213,155)(10,106,184,156,40,76,214,126)(11,77,185,127,41,107,215,157)(12,108,186,158,42,78,216,128)(13,79,187,129,43,109,217,159)(14,110,188,160,44,80,218,130)(15,81,189,131,45,111,219,161)(16,112,190,162,46,82,220,132)(17,83,191,133,47,113,221,163)(18,114,192,164,48,84,222,134)(19,85,193,135,49,115,223,165)(20,116,194,166,50,86,224,136)(21,87,195,137,51,117,225,167)(22,118,196,168,52,88,226,138)(23,89,197,139,53,119,227,169)(24,120,198,170,54,90,228,140)(25,91,199,141,55,61,229,171)(26,62,200,172,56,92,230,142)(27,93,201,143,57,63,231,173)(28,64,202,174,58,94,232,144)(29,95,203,145,59,65,233,175)(30,66,204,176,60,96,234,146)>;
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)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(173,180)(174,179)(175,178)(176,177)(181,228)(182,227)(183,226)(184,225)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)(200,209)(201,208)(202,207)(203,206)(204,205)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,67,235,177,31,97,205,147)(2,98,236,148,32,68,206,178)(3,69,237,179,33,99,207,149)(4,100,238,150,34,70,208,180)(5,71,239,121,35,101,209,151)(6,102,240,152,36,72,210,122)(7,73,181,123,37,103,211,153)(8,104,182,154,38,74,212,124)(9,75,183,125,39,105,213,155)(10,106,184,156,40,76,214,126)(11,77,185,127,41,107,215,157)(12,108,186,158,42,78,216,128)(13,79,187,129,43,109,217,159)(14,110,188,160,44,80,218,130)(15,81,189,131,45,111,219,161)(16,112,190,162,46,82,220,132)(17,83,191,133,47,113,221,163)(18,114,192,164,48,84,222,134)(19,85,193,135,49,115,223,165)(20,116,194,166,50,86,224,136)(21,87,195,137,51,117,225,167)(22,118,196,168,52,88,226,138)(23,89,197,139,53,119,227,169)(24,120,198,170,54,90,228,140)(25,91,199,141,55,61,229,171)(26,62,200,172,56,92,230,142)(27,93,201,143,57,63,231,173)(28,64,202,174,58,94,232,144)(29,95,203,145,59,65,233,175)(30,66,204,176,60,96,234,146) );
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),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(121,172),(122,171),(123,170),(124,169),(125,168),(126,167),(127,166),(128,165),(129,164),(130,163),(131,162),(132,161),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(173,180),(174,179),(175,178),(176,177),(181,228),(182,227),(183,226),(184,225),(185,224),(186,223),(187,222),(188,221),(189,220),(190,219),(191,218),(192,217),(193,216),(194,215),(195,214),(196,213),(197,212),(198,211),(199,210),(200,209),(201,208),(202,207),(203,206),(204,205),(229,240),(230,239),(231,238),(232,237),(233,236),(234,235)], [(1,67,235,177,31,97,205,147),(2,98,236,148,32,68,206,178),(3,69,237,179,33,99,207,149),(4,100,238,150,34,70,208,180),(5,71,239,121,35,101,209,151),(6,102,240,152,36,72,210,122),(7,73,181,123,37,103,211,153),(8,104,182,154,38,74,212,124),(9,75,183,125,39,105,213,155),(10,106,184,156,40,76,214,126),(11,77,185,127,41,107,215,157),(12,108,186,158,42,78,216,128),(13,79,187,129,43,109,217,159),(14,110,188,160,44,80,218,130),(15,81,189,131,45,111,219,161),(16,112,190,162,46,82,220,132),(17,83,191,133,47,113,221,163),(18,114,192,164,48,84,222,134),(19,85,193,135,49,115,223,165),(20,116,194,166,50,86,224,136),(21,87,195,137,51,117,225,167),(22,118,196,168,52,88,226,138),(23,89,197,139,53,119,227,169),(24,120,198,170,54,90,228,140),(25,91,199,141,55,61,229,171),(26,62,200,172,56,92,230,142),(27,93,201,143,57,63,231,173),(28,64,202,174,58,94,232,144),(29,95,203,145,59,65,233,175),(30,66,204,176,60,96,234,146)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 30 | 30 | 2 | 1 | 1 | 2 | 30 | 30 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×S3 | D15 | C8○D4 | C4×D5 | C4×D5 | D30 | D30 | C4×D15 | C4×D15 | D12.C4 | D20.2C4 | D60.3C4 |
| kernel | D60.3C4 | C8×D15 | C40⋊S3 | C2×C15⋊3C8 | C15×M4(2) | D60⋊11C2 | Dic30 | D60 | C15⋊7D4 | C5×M4(2) | C3×M4(2) | C40 | C2×C20 | C24 | C2×C12 | C20 | C2×C10 | M4(2) | C15 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of D60.3C4 ►in GL4(𝔽241) generated by
| 16 | 131 | 0 | 0 |
| 46 | 211 | 0 | 0 |
| 0 | 0 | 64 | 128 |
| 0 | 0 | 0 | 177 |
| 178 | 147 | 0 | 0 |
| 114 | 63 | 0 | 0 |
| 0 | 0 | 64 | 128 |
| 0 | 0 | 177 | 177 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 211 | 211 |
G:=sub<GL(4,GF(241))| [16,46,0,0,131,211,0,0,0,0,64,0,0,0,128,177],[178,114,0,0,147,63,0,0,0,0,64,177,0,0,128,177],[64,0,0,0,0,64,0,0,0,0,30,211,0,0,0,211] >;
D60.3C4 in GAP, Magma, Sage, TeX
D_{60}._3C_4 % in TeX
G:=Group("D60.3C4"); // GroupNames label
G:=SmallGroup(480,872);
// by ID
G=gap.SmallGroup(480,872);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,80,2693,18822]);
// 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^31,c*b*c^-1=a^30*b>;
// generators/relations