metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊12C4, C6.11D40, C60.33D4, C30.17D8, C20.49D12, C30.5SD16, C4⋊Dic5⋊1S3, C12.3(C4×D5), C20.63(C4×S3), C60.91(C2×C4), (C2×C30).15D4, (C2×C6).30D20, C5⋊2(C6.D8), C15⋊7(D4⋊C4), C3⋊1(D20⋊5C4), C10.6(D4⋊S3), (C2×D60).12C2, (C2×C12).50D10, (C2×C20).280D6, C6.4(C40⋊C2), C2.2(C3⋊D40), C10.19(D6⋊C4), C4.14(C5⋊D12), C12.13(C5⋊D4), C4.1(D30.C2), (C2×C60).99C22, C6.4(D10⋊C4), C2.5(D30⋊4C4), C30.51(C22⋊C4), C10.1(Q8⋊2S3), C2.1(C15⋊SD16), C22.13(C3⋊D20), (C2×C3⋊C8)⋊2D5, (C10×C3⋊C8)⋊5C2, (C3×C4⋊Dic5)⋊1C2, (C2×C4).83(S3×D5), (C2×C10).24(C3⋊D4), SmallGroup(480,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊12C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a33b >
Subgroups: 812 in 100 conjugacy classes, 40 normal (38 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, Dic5, C20, D10, C2×C10, C3⋊C8, D12, C2×C12, C2×C12, C22×S3, D15, C30, D4⋊C4, C40, D20, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60, D30, C2×C30, C4⋊Dic5, C2×C40, C2×D20, C6.D8, C5×C3⋊C8, C6×Dic5, D60, D60, C2×C60, C22×D15, D20⋊5C4, C3×C4⋊Dic5, C10×C3⋊C8, C2×D60, D60⋊12C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D4⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D4⋊S3, Q8⋊2S3, S3×D5, C40⋊C2, D40, D10⋊C4, C6.D8, D30.C2, C3⋊D20, C5⋊D12, D20⋊5C4, C3⋊D40, C15⋊SD16, D30⋊4C4, D60⋊12C4
►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)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(115 120)(116 119)(117 118)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(152 180)(153 179)(154 178)(155 177)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(181 211)(182 210)(183 209)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(212 240)(213 239)(214 238)(215 237)(216 236)(217 235)(218 234)(219 233)(220 232)(221 231)(222 230)(223 229)(224 228)(225 227)
(1 219 88 129)(2 238 89 148)(3 197 90 167)(4 216 91 126)(5 235 92 145)(6 194 93 164)(7 213 94 123)(8 232 95 142)(9 191 96 161)(10 210 97 180)(11 229 98 139)(12 188 99 158)(13 207 100 177)(14 226 101 136)(15 185 102 155)(16 204 103 174)(17 223 104 133)(18 182 105 152)(19 201 106 171)(20 220 107 130)(21 239 108 149)(22 198 109 168)(23 217 110 127)(24 236 111 146)(25 195 112 165)(26 214 113 124)(27 233 114 143)(28 192 115 162)(29 211 116 121)(30 230 117 140)(31 189 118 159)(32 208 119 178)(33 227 120 137)(34 186 61 156)(35 205 62 175)(36 224 63 134)(37 183 64 153)(38 202 65 172)(39 221 66 131)(40 240 67 150)(41 199 68 169)(42 218 69 128)(43 237 70 147)(44 196 71 166)(45 215 72 125)(46 234 73 144)(47 193 74 163)(48 212 75 122)(49 231 76 141)(50 190 77 160)(51 209 78 179)(52 228 79 138)(53 187 80 157)(54 206 81 176)(55 225 82 135)(56 184 83 154)(57 203 84 173)(58 222 85 132)(59 181 86 151)(60 200 87 170)
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)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(115,120)(116,119)(117,118)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(181,211)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(212,240)(213,239)(214,238)(215,237)(216,236)(217,235)(218,234)(219,233)(220,232)(221,231)(222,230)(223,229)(224,228)(225,227), (1,219,88,129)(2,238,89,148)(3,197,90,167)(4,216,91,126)(5,235,92,145)(6,194,93,164)(7,213,94,123)(8,232,95,142)(9,191,96,161)(10,210,97,180)(11,229,98,139)(12,188,99,158)(13,207,100,177)(14,226,101,136)(15,185,102,155)(16,204,103,174)(17,223,104,133)(18,182,105,152)(19,201,106,171)(20,220,107,130)(21,239,108,149)(22,198,109,168)(23,217,110,127)(24,236,111,146)(25,195,112,165)(26,214,113,124)(27,233,114,143)(28,192,115,162)(29,211,116,121)(30,230,117,140)(31,189,118,159)(32,208,119,178)(33,227,120,137)(34,186,61,156)(35,205,62,175)(36,224,63,134)(37,183,64,153)(38,202,65,172)(39,221,66,131)(40,240,67,150)(41,199,68,169)(42,218,69,128)(43,237,70,147)(44,196,71,166)(45,215,72,125)(46,234,73,144)(47,193,74,163)(48,212,75,122)(49,231,76,141)(50,190,77,160)(51,209,78,179)(52,228,79,138)(53,187,80,157)(54,206,81,176)(55,225,82,135)(56,184,83,154)(57,203,84,173)(58,222,85,132)(59,181,86,151)(60,200,87,170)>;
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)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(115,120)(116,119)(117,118)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(181,211)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(212,240)(213,239)(214,238)(215,237)(216,236)(217,235)(218,234)(219,233)(220,232)(221,231)(222,230)(223,229)(224,228)(225,227), (1,219,88,129)(2,238,89,148)(3,197,90,167)(4,216,91,126)(5,235,92,145)(6,194,93,164)(7,213,94,123)(8,232,95,142)(9,191,96,161)(10,210,97,180)(11,229,98,139)(12,188,99,158)(13,207,100,177)(14,226,101,136)(15,185,102,155)(16,204,103,174)(17,223,104,133)(18,182,105,152)(19,201,106,171)(20,220,107,130)(21,239,108,149)(22,198,109,168)(23,217,110,127)(24,236,111,146)(25,195,112,165)(26,214,113,124)(27,233,114,143)(28,192,115,162)(29,211,116,121)(30,230,117,140)(31,189,118,159)(32,208,119,178)(33,227,120,137)(34,186,61,156)(35,205,62,175)(36,224,63,134)(37,183,64,153)(38,202,65,172)(39,221,66,131)(40,240,67,150)(41,199,68,169)(42,218,69,128)(43,237,70,147)(44,196,71,166)(45,215,72,125)(46,234,73,144)(47,193,74,163)(48,212,75,122)(49,231,76,141)(50,190,77,160)(51,209,78,179)(52,228,79,138)(53,187,80,157)(54,206,81,176)(55,225,82,135)(56,184,83,154)(57,203,84,173)(58,222,85,132)(59,181,86,151)(60,200,87,170) );
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),(61,114),(62,113),(63,112),(64,111),(65,110),(66,109),(67,108),(68,107),(69,106),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(115,120),(116,119),(117,118),(121,151),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139),(134,138),(135,137),(152,180),(153,179),(154,178),(155,177),(156,176),(157,175),(158,174),(159,173),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167),(181,211),(182,210),(183,209),(184,208),(185,207),(186,206),(187,205),(188,204),(189,203),(190,202),(191,201),(192,200),(193,199),(194,198),(195,197),(212,240),(213,239),(214,238),(215,237),(216,236),(217,235),(218,234),(219,233),(220,232),(221,231),(222,230),(223,229),(224,228),(225,227)], [(1,219,88,129),(2,238,89,148),(3,197,90,167),(4,216,91,126),(5,235,92,145),(6,194,93,164),(7,213,94,123),(8,232,95,142),(9,191,96,161),(10,210,97,180),(11,229,98,139),(12,188,99,158),(13,207,100,177),(14,226,101,136),(15,185,102,155),(16,204,103,174),(17,223,104,133),(18,182,105,152),(19,201,106,171),(20,220,107,130),(21,239,108,149),(22,198,109,168),(23,217,110,127),(24,236,111,146),(25,195,112,165),(26,214,113,124),(27,233,114,143),(28,192,115,162),(29,211,116,121),(30,230,117,140),(31,189,118,159),(32,208,119,178),(33,227,120,137),(34,186,61,156),(35,205,62,175),(36,224,63,134),(37,183,64,153),(38,202,65,172),(39,221,66,131),(40,240,67,150),(41,199,68,169),(42,218,69,128),(43,237,70,147),(44,196,71,166),(45,215,72,125),(46,234,73,144),(47,193,74,163),(48,212,75,122),(49,231,76,141),(50,190,77,160),(51,209,78,179),(52,228,79,138),(53,187,80,157),(54,206,81,176),(55,225,82,135),(56,184,83,154),(57,203,84,173),(58,222,85,132),(59,181,86,151),(60,200,87,170)]])
72 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 | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | D10 | C4×S3 | D12 | C3⋊D4 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 | D4⋊S3 | Q8⋊2S3 | S3×D5 | D30.C2 | C5⋊D12 | C3⋊D20 | C3⋊D40 | C15⋊SD16 |
| kernel | D60⋊12C4 | C3×C4⋊Dic5 | C10×C3⋊C8 | C2×D60 | D60 | C4⋊Dic5 | C60 | C2×C30 | C2×C3⋊C8 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C6 | C6 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of D60⋊12C4 ►in GL6(𝔽241)
| 220 | 50 | 0 | 0 | 0 | 0 |
| 49 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 189 | 0 | 0 |
| 0 | 0 | 52 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 187 |
| 0 | 0 | 0 | 0 | 67 | 239 |
| 21 | 68 | 0 | 0 | 0 | 0 |
| 192 | 220 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 187 |
| 0 | 0 | 0 | 0 | 67 | 239 |
| 102 | 67 | 0 | 0 | 0 | 0 |
| 208 | 139 | 0 | 0 | 0 | 0 |
| 0 | 0 | 195 | 46 | 0 | 0 |
| 0 | 0 | 64 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [220,49,0,0,0,0,50,21,0,0,0,0,0,0,52,52,0,0,0,0,189,240,0,0,0,0,0,0,1,67,0,0,0,0,187,239],[21,192,0,0,0,0,68,220,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,67,0,0,0,0,187,239],[102,208,0,0,0,0,67,139,0,0,0,0,0,0,195,64,0,0,0,0,46,46,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
D60⋊12C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{12}C_4 % in TeX
G:=Group("D60:12C4"); // GroupNames label
G:=SmallGroup(480,44);
// by ID
G=gap.SmallGroup(480,44);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,204,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^33*b>;
// generators/relations