direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D60, C20⋊5D12, C60⋊23D4, C12⋊5D20, C42⋊5D15, C3⋊3(C4×D20), C5⋊4(C4×D12), (C4×C20)⋊9S3, (C4×C60)⋊9C2, (C4×C12)⋊7D5, C4⋊2(C4×D15), C15⋊31(C4×D4), C20⋊14(C4×S3), C60⋊32(C2×C4), C12⋊10(C4×D5), C2.1(C2×D60), D30⋊18(C2×C4), C60⋊5C4⋊34C2, C6.30(C2×D20), (C2×C4).77D30, (C2×D60).20C2, C10.31(C2×D12), (C2×C20).412D6, C30.259(C2×D4), D30⋊3C4⋊39C2, C6.92(C4○D20), (C2×C12).393D10, C30.166(C4○D4), C10.92(C4○D12), C30.155(C22×C4), (C2×C30).270C23, (C2×C60).460C22, C2.3(D60⋊11C2), C22.11(C22×D15), (C22×D15).78C22, (C2×Dic15).157C22, C6.60(C2×C4×D5), C2.6(C2×C4×D15), (C2×C4×D15)⋊13C2, C10.92(S3×C2×C4), (C2×C6).266(C22×D5), (C2×C10).265(C22×S3), SmallGroup(480,838)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D60
G = < a,b,c | a4=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 1188 in 188 conjugacy classes, 71 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, Dic15, C60, C60, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, C4×D12, C4×D15, D60, C2×Dic15, C2×C60, C22×D15, C4×D20, C60⋊5C4, D30⋊3C4, C4×C60, C2×C4×D15, C2×D60, C4×D60
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, D12, C22×S3, D15, C4×D4, C4×D5, D20, C22×D5, S3×C2×C4, C2×D12, C4○D12, D30, C2×C4×D5, C2×D20, C4○D20, C4×D12, C4×D15, D60, C22×D15, C4×D20, C2×C4×D15, C2×D60, D60⋊11C2, C4×D60
►On 240 points
Generators in S240
(1 185 179 106)(2 186 180 107)(3 187 121 108)(4 188 122 109)(5 189 123 110)(6 190 124 111)(7 191 125 112)(8 192 126 113)(9 193 127 114)(10 194 128 115)(11 195 129 116)(12 196 130 117)(13 197 131 118)(14 198 132 119)(15 199 133 120)(16 200 134 61)(17 201 135 62)(18 202 136 63)(19 203 137 64)(20 204 138 65)(21 205 139 66)(22 206 140 67)(23 207 141 68)(24 208 142 69)(25 209 143 70)(26 210 144 71)(27 211 145 72)(28 212 146 73)(29 213 147 74)(30 214 148 75)(31 215 149 76)(32 216 150 77)(33 217 151 78)(34 218 152 79)(35 219 153 80)(36 220 154 81)(37 221 155 82)(38 222 156 83)(39 223 157 84)(40 224 158 85)(41 225 159 86)(42 226 160 87)(43 227 161 88)(44 228 162 89)(45 229 163 90)(46 230 164 91)(47 231 165 92)(48 232 166 93)(49 233 167 94)(50 234 168 95)(51 235 169 96)(52 236 170 97)(53 237 171 98)(54 238 172 99)(55 239 173 100)(56 240 174 101)(57 181 175 102)(58 182 176 103)(59 183 177 104)(60 184 178 105)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 180)(15 179)(16 178)(17 177)(18 176)(19 175)(20 174)(21 173)(22 172)(23 171)(24 170)(25 169)(26 168)(27 167)(28 166)(29 165)(30 164)(31 163)(32 162)(33 161)(34 160)(35 159)(36 158)(37 157)(38 156)(39 155)(40 154)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 140)(55 139)(56 138)(57 137)(58 136)(59 135)(60 134)(61 184)(62 183)(63 182)(64 181)(65 240)(66 239)(67 238)(68 237)(69 236)(70 235)(71 234)(72 233)(73 232)(74 231)(75 230)(76 229)(77 228)(78 227)(79 226)(80 225)(81 224)(82 223)(83 222)(84 221)(85 220)(86 219)(87 218)(88 217)(89 216)(90 215)(91 214)(92 213)(93 212)(94 211)(95 210)(96 209)(97 208)(98 207)(99 206)(100 205)(101 204)(102 203)(103 202)(104 201)(105 200)(106 199)(107 198)(108 197)(109 196)(110 195)(111 194)(112 193)(113 192)(114 191)(115 190)(116 189)(117 188)(118 187)(119 186)(120 185)
G:=sub<Sym(240)| (1,185,179,106)(2,186,180,107)(3,187,121,108)(4,188,122,109)(5,189,123,110)(6,190,124,111)(7,191,125,112)(8,192,126,113)(9,193,127,114)(10,194,128,115)(11,195,129,116)(12,196,130,117)(13,197,131,118)(14,198,132,119)(15,199,133,120)(16,200,134,61)(17,201,135,62)(18,202,136,63)(19,203,137,64)(20,204,138,65)(21,205,139,66)(22,206,140,67)(23,207,141,68)(24,208,142,69)(25,209,143,70)(26,210,144,71)(27,211,145,72)(28,212,146,73)(29,213,147,74)(30,214,148,75)(31,215,149,76)(32,216,150,77)(33,217,151,78)(34,218,152,79)(35,219,153,80)(36,220,154,81)(37,221,155,82)(38,222,156,83)(39,223,157,84)(40,224,158,85)(41,225,159,86)(42,226,160,87)(43,227,161,88)(44,228,162,89)(45,229,163,90)(46,230,164,91)(47,231,165,92)(48,232,166,93)(49,233,167,94)(50,234,168,95)(51,235,169,96)(52,236,170,97)(53,237,171,98)(54,238,172,99)(55,239,173,100)(56,240,174,101)(57,181,175,102)(58,182,176,103)(59,183,177,104)(60,184,178,105), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,180)(15,179)(16,178)(17,177)(18,176)(19,175)(20,174)(21,173)(22,172)(23,171)(24,170)(25,169)(26,168)(27,167)(28,166)(29,165)(30,164)(31,163)(32,162)(33,161)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,140)(55,139)(56,138)(57,137)(58,136)(59,135)(60,134)(61,184)(62,183)(63,182)(64,181)(65,240)(66,239)(67,238)(68,237)(69,236)(70,235)(71,234)(72,233)(73,232)(74,231)(75,230)(76,229)(77,228)(78,227)(79,226)(80,225)(81,224)(82,223)(83,222)(84,221)(85,220)(86,219)(87,218)(88,217)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)(117,188)(118,187)(119,186)(120,185)>;
G:=Group( (1,185,179,106)(2,186,180,107)(3,187,121,108)(4,188,122,109)(5,189,123,110)(6,190,124,111)(7,191,125,112)(8,192,126,113)(9,193,127,114)(10,194,128,115)(11,195,129,116)(12,196,130,117)(13,197,131,118)(14,198,132,119)(15,199,133,120)(16,200,134,61)(17,201,135,62)(18,202,136,63)(19,203,137,64)(20,204,138,65)(21,205,139,66)(22,206,140,67)(23,207,141,68)(24,208,142,69)(25,209,143,70)(26,210,144,71)(27,211,145,72)(28,212,146,73)(29,213,147,74)(30,214,148,75)(31,215,149,76)(32,216,150,77)(33,217,151,78)(34,218,152,79)(35,219,153,80)(36,220,154,81)(37,221,155,82)(38,222,156,83)(39,223,157,84)(40,224,158,85)(41,225,159,86)(42,226,160,87)(43,227,161,88)(44,228,162,89)(45,229,163,90)(46,230,164,91)(47,231,165,92)(48,232,166,93)(49,233,167,94)(50,234,168,95)(51,235,169,96)(52,236,170,97)(53,237,171,98)(54,238,172,99)(55,239,173,100)(56,240,174,101)(57,181,175,102)(58,182,176,103)(59,183,177,104)(60,184,178,105), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,180)(15,179)(16,178)(17,177)(18,176)(19,175)(20,174)(21,173)(22,172)(23,171)(24,170)(25,169)(26,168)(27,167)(28,166)(29,165)(30,164)(31,163)(32,162)(33,161)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,140)(55,139)(56,138)(57,137)(58,136)(59,135)(60,134)(61,184)(62,183)(63,182)(64,181)(65,240)(66,239)(67,238)(68,237)(69,236)(70,235)(71,234)(72,233)(73,232)(74,231)(75,230)(76,229)(77,228)(78,227)(79,226)(80,225)(81,224)(82,223)(83,222)(84,221)(85,220)(86,219)(87,218)(88,217)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)(117,188)(118,187)(119,186)(120,185) );
G=PermutationGroup([[(1,185,179,106),(2,186,180,107),(3,187,121,108),(4,188,122,109),(5,189,123,110),(6,190,124,111),(7,191,125,112),(8,192,126,113),(9,193,127,114),(10,194,128,115),(11,195,129,116),(12,196,130,117),(13,197,131,118),(14,198,132,119),(15,199,133,120),(16,200,134,61),(17,201,135,62),(18,202,136,63),(19,203,137,64),(20,204,138,65),(21,205,139,66),(22,206,140,67),(23,207,141,68),(24,208,142,69),(25,209,143,70),(26,210,144,71),(27,211,145,72),(28,212,146,73),(29,213,147,74),(30,214,148,75),(31,215,149,76),(32,216,150,77),(33,217,151,78),(34,218,152,79),(35,219,153,80),(36,220,154,81),(37,221,155,82),(38,222,156,83),(39,223,157,84),(40,224,158,85),(41,225,159,86),(42,226,160,87),(43,227,161,88),(44,228,162,89),(45,229,163,90),(46,230,164,91),(47,231,165,92),(48,232,166,93),(49,233,167,94),(50,234,168,95),(51,235,169,96),(52,236,170,97),(53,237,171,98),(54,238,172,99),(55,239,173,100),(56,240,174,101),(57,181,175,102),(58,182,176,103),(59,183,177,104),(60,184,178,105)], [(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)], [(1,133),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,180),(15,179),(16,178),(17,177),(18,176),(19,175),(20,174),(21,173),(22,172),(23,171),(24,170),(25,169),(26,168),(27,167),(28,166),(29,165),(30,164),(31,163),(32,162),(33,161),(34,160),(35,159),(36,158),(37,157),(38,156),(39,155),(40,154),(41,153),(42,152),(43,151),(44,150),(45,149),(46,148),(47,147),(48,146),(49,145),(50,144),(51,143),(52,142),(53,141),(54,140),(55,139),(56,138),(57,137),(58,136),(59,135),(60,134),(61,184),(62,183),(63,182),(64,181),(65,240),(66,239),(67,238),(68,237),(69,236),(70,235),(71,234),(72,233),(73,232),(74,231),(75,230),(76,229),(77,228),(78,227),(79,226),(80,225),(81,224),(82,223),(83,222),(84,221),(85,220),(86,219),(87,218),(88,217),(89,216),(90,215),(91,214),(92,213),(93,212),(94,211),(95,210),(96,209),(97,208),(98,207),(99,206),(100,205),(101,204),(102,203),(103,202),(104,201),(105,200),(106,199),(107,198),(108,197),(109,196),(110,195),(111,194),(112,193),(113,192),(114,191),(115,190),(116,189),(117,188),(118,187),(119,186),(120,185)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 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 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | C4○D4 | D10 | C4×S3 | D12 | D15 | C4×D5 | D20 | C4○D12 | D30 | C4○D20 | C4×D15 | D60 | D60⋊11C2 |
| kernel | C4×D60 | C60⋊5C4 | D30⋊3C4 | C4×C60 | C2×C4×D15 | C2×D60 | D60 | C4×C20 | C60 | C4×C12 | C2×C20 | C30 | C2×C12 | C20 | C20 | C42 | C12 | C12 | C10 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 4 | 8 | 8 | 4 | 12 | 8 | 16 | 16 | 16 |
Matrix representation of C4×D60 ►in GL3(𝔽61) generated by
| 50 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 60 | 0 | 0 |
| 0 | 2 | 8 |
| 0 | 31 | 33 |
| 1 | 0 | 0 |
| 0 | 53 | 28 |
| 0 | 13 | 8 |
G:=sub<GL(3,GF(61))| [50,0,0,0,11,0,0,0,11],[60,0,0,0,2,31,0,8,33],[1,0,0,0,53,13,0,28,8] >;
C4×D60 in GAP, Magma, Sage, TeX
C_4\times D_{60} % in TeX
G:=Group("C4xD60"); // GroupNames label
G:=SmallGroup(480,838);
// by ID
G=gap.SmallGroup(480,838);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations