metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4.2D5, C60⋊5C4⋊2C2, (C2×C20).7D6, (C2×C12).7D10, C6.4(C4○D20), C10.D4⋊4S3, D6⋊Dic5.3C2, (C2×C60).3C22, Dic15⋊5C4⋊1C2, (C2×Dic5).5D6, C15⋊3(C42⋊2C2), (Dic3×Dic5)⋊5C2, C30.11(C4○D4), (C2×C30).32C23, (C22×S3).3D10, C10.49(C4○D12), C6.18(D4⋊2D5), C2.7(D12⋊5D5), C2.8(D60⋊C2), C10.5(Q8⋊3S3), (C2×Dic3).82D10, C3⋊2(C23.D10), C10.63(D4⋊2S3), C2.9(C30.C23), (C6×Dic5).17C22, (C2×Dic15).39C22, (C10×Dic3).16C22, C5⋊5(C4⋊C4⋊S3), (C5×D6⋊C4).2C2, (C2×C4).29(S3×D5), (S3×C2×C10).3C22, C22.123(C2×S3×D5), (C3×C10.D4)⋊4C2, (C2×C6).44(C22×D5), (C2×C10).44(C22×S3), SmallGroup(480,418)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊5C4⋊C2
G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a41b2, cbc=a30b >
Subgroups: 524 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C4⋊C4⋊S3, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C23.D10, Dic3×Dic5, D6⋊Dic5, Dic15⋊5C4, C3×C10.D4, C5×D6⋊C4, C60⋊5C4, C60⋊5C4⋊C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, Q8⋊3S3, S3×D5, C4○D20, D4⋊2D5, C4⋊C4⋊S3, C2×S3×D5, C23.D10, D60⋊C2, D12⋊5D5, C30.C23, C60⋊5C4⋊C2
►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 99 156 193)(2 98 157 192)(3 97 158 191)(4 96 159 190)(5 95 160 189)(6 94 161 188)(7 93 162 187)(8 92 163 186)(9 91 164 185)(10 90 165 184)(11 89 166 183)(12 88 167 182)(13 87 168 181)(14 86 169 240)(15 85 170 239)(16 84 171 238)(17 83 172 237)(18 82 173 236)(19 81 174 235)(20 80 175 234)(21 79 176 233)(22 78 177 232)(23 77 178 231)(24 76 179 230)(25 75 180 229)(26 74 121 228)(27 73 122 227)(28 72 123 226)(29 71 124 225)(30 70 125 224)(31 69 126 223)(32 68 127 222)(33 67 128 221)(34 66 129 220)(35 65 130 219)(36 64 131 218)(37 63 132 217)(38 62 133 216)(39 61 134 215)(40 120 135 214)(41 119 136 213)(42 118 137 212)(43 117 138 211)(44 116 139 210)(45 115 140 209)(46 114 141 208)(47 113 142 207)(48 112 143 206)(49 111 144 205)(50 110 145 204)(51 109 146 203)(52 108 147 202)(53 107 148 201)(54 106 149 200)(55 105 150 199)(56 104 151 198)(57 103 152 197)(58 102 153 196)(59 101 154 195)(60 100 155 194)
(2 137)(3 23)(4 159)(5 45)(6 121)(8 143)(9 29)(10 165)(11 51)(12 127)(14 149)(15 35)(16 171)(17 57)(18 133)(20 155)(21 41)(22 177)(24 139)(26 161)(27 47)(28 123)(30 145)(32 167)(33 53)(34 129)(36 151)(38 173)(39 59)(40 135)(42 157)(44 179)(46 141)(48 163)(50 125)(52 147)(54 169)(56 131)(58 153)(60 175)(61 71)(62 206)(63 93)(64 228)(65 115)(66 190)(67 77)(68 212)(69 99)(70 234)(72 196)(73 83)(74 218)(75 105)(76 240)(78 202)(79 89)(80 224)(81 111)(82 186)(84 208)(85 95)(86 230)(87 117)(88 192)(90 214)(91 101)(92 236)(94 198)(96 220)(97 107)(98 182)(100 204)(102 226)(103 113)(104 188)(106 210)(108 232)(109 119)(110 194)(112 216)(114 238)(116 200)(118 222)(120 184)(122 142)(124 164)(128 148)(130 170)(134 154)(136 176)(140 160)(146 166)(152 172)(158 178)(181 211)(183 233)(185 195)(187 217)(189 239)(191 201)(193 223)(197 207)(199 229)(203 213)(205 235)(209 219)(215 225)(221 231)(227 237)
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), (1,99,156,193)(2,98,157,192)(3,97,158,191)(4,96,159,190)(5,95,160,189)(6,94,161,188)(7,93,162,187)(8,92,163,186)(9,91,164,185)(10,90,165,184)(11,89,166,183)(12,88,167,182)(13,87,168,181)(14,86,169,240)(15,85,170,239)(16,84,171,238)(17,83,172,237)(18,82,173,236)(19,81,174,235)(20,80,175,234)(21,79,176,233)(22,78,177,232)(23,77,178,231)(24,76,179,230)(25,75,180,229)(26,74,121,228)(27,73,122,227)(28,72,123,226)(29,71,124,225)(30,70,125,224)(31,69,126,223)(32,68,127,222)(33,67,128,221)(34,66,129,220)(35,65,130,219)(36,64,131,218)(37,63,132,217)(38,62,133,216)(39,61,134,215)(40,120,135,214)(41,119,136,213)(42,118,137,212)(43,117,138,211)(44,116,139,210)(45,115,140,209)(46,114,141,208)(47,113,142,207)(48,112,143,206)(49,111,144,205)(50,110,145,204)(51,109,146,203)(52,108,147,202)(53,107,148,201)(54,106,149,200)(55,105,150,199)(56,104,151,198)(57,103,152,197)(58,102,153,196)(59,101,154,195)(60,100,155,194), (2,137)(3,23)(4,159)(5,45)(6,121)(8,143)(9,29)(10,165)(11,51)(12,127)(14,149)(15,35)(16,171)(17,57)(18,133)(20,155)(21,41)(22,177)(24,139)(26,161)(27,47)(28,123)(30,145)(32,167)(33,53)(34,129)(36,151)(38,173)(39,59)(40,135)(42,157)(44,179)(46,141)(48,163)(50,125)(52,147)(54,169)(56,131)(58,153)(60,175)(61,71)(62,206)(63,93)(64,228)(65,115)(66,190)(67,77)(68,212)(69,99)(70,234)(72,196)(73,83)(74,218)(75,105)(76,240)(78,202)(79,89)(80,224)(81,111)(82,186)(84,208)(85,95)(86,230)(87,117)(88,192)(90,214)(91,101)(92,236)(94,198)(96,220)(97,107)(98,182)(100,204)(102,226)(103,113)(104,188)(106,210)(108,232)(109,119)(110,194)(112,216)(114,238)(116,200)(118,222)(120,184)(122,142)(124,164)(128,148)(130,170)(134,154)(136,176)(140,160)(146,166)(152,172)(158,178)(181,211)(183,233)(185,195)(187,217)(189,239)(191,201)(193,223)(197,207)(199,229)(203,213)(205,235)(209,219)(215,225)(221,231)(227,237)>;
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), (1,99,156,193)(2,98,157,192)(3,97,158,191)(4,96,159,190)(5,95,160,189)(6,94,161,188)(7,93,162,187)(8,92,163,186)(9,91,164,185)(10,90,165,184)(11,89,166,183)(12,88,167,182)(13,87,168,181)(14,86,169,240)(15,85,170,239)(16,84,171,238)(17,83,172,237)(18,82,173,236)(19,81,174,235)(20,80,175,234)(21,79,176,233)(22,78,177,232)(23,77,178,231)(24,76,179,230)(25,75,180,229)(26,74,121,228)(27,73,122,227)(28,72,123,226)(29,71,124,225)(30,70,125,224)(31,69,126,223)(32,68,127,222)(33,67,128,221)(34,66,129,220)(35,65,130,219)(36,64,131,218)(37,63,132,217)(38,62,133,216)(39,61,134,215)(40,120,135,214)(41,119,136,213)(42,118,137,212)(43,117,138,211)(44,116,139,210)(45,115,140,209)(46,114,141,208)(47,113,142,207)(48,112,143,206)(49,111,144,205)(50,110,145,204)(51,109,146,203)(52,108,147,202)(53,107,148,201)(54,106,149,200)(55,105,150,199)(56,104,151,198)(57,103,152,197)(58,102,153,196)(59,101,154,195)(60,100,155,194), (2,137)(3,23)(4,159)(5,45)(6,121)(8,143)(9,29)(10,165)(11,51)(12,127)(14,149)(15,35)(16,171)(17,57)(18,133)(20,155)(21,41)(22,177)(24,139)(26,161)(27,47)(28,123)(30,145)(32,167)(33,53)(34,129)(36,151)(38,173)(39,59)(40,135)(42,157)(44,179)(46,141)(48,163)(50,125)(52,147)(54,169)(56,131)(58,153)(60,175)(61,71)(62,206)(63,93)(64,228)(65,115)(66,190)(67,77)(68,212)(69,99)(70,234)(72,196)(73,83)(74,218)(75,105)(76,240)(78,202)(79,89)(80,224)(81,111)(82,186)(84,208)(85,95)(86,230)(87,117)(88,192)(90,214)(91,101)(92,236)(94,198)(96,220)(97,107)(98,182)(100,204)(102,226)(103,113)(104,188)(106,210)(108,232)(109,119)(110,194)(112,216)(114,238)(116,200)(118,222)(120,184)(122,142)(124,164)(128,148)(130,170)(134,154)(136,176)(140,160)(146,166)(152,172)(158,178)(181,211)(183,233)(185,195)(187,217)(189,239)(191,201)(193,223)(197,207)(199,229)(203,213)(205,235)(209,219)(215,225)(221,231)(227,237) );
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)], [(1,99,156,193),(2,98,157,192),(3,97,158,191),(4,96,159,190),(5,95,160,189),(6,94,161,188),(7,93,162,187),(8,92,163,186),(9,91,164,185),(10,90,165,184),(11,89,166,183),(12,88,167,182),(13,87,168,181),(14,86,169,240),(15,85,170,239),(16,84,171,238),(17,83,172,237),(18,82,173,236),(19,81,174,235),(20,80,175,234),(21,79,176,233),(22,78,177,232),(23,77,178,231),(24,76,179,230),(25,75,180,229),(26,74,121,228),(27,73,122,227),(28,72,123,226),(29,71,124,225),(30,70,125,224),(31,69,126,223),(32,68,127,222),(33,67,128,221),(34,66,129,220),(35,65,130,219),(36,64,131,218),(37,63,132,217),(38,62,133,216),(39,61,134,215),(40,120,135,214),(41,119,136,213),(42,118,137,212),(43,117,138,211),(44,116,139,210),(45,115,140,209),(46,114,141,208),(47,113,142,207),(48,112,143,206),(49,111,144,205),(50,110,145,204),(51,109,146,203),(52,108,147,202),(53,107,148,201),(54,106,149,200),(55,105,150,199),(56,104,151,198),(57,103,152,197),(58,102,153,196),(59,101,154,195),(60,100,155,194)], [(2,137),(3,23),(4,159),(5,45),(6,121),(8,143),(9,29),(10,165),(11,51),(12,127),(14,149),(15,35),(16,171),(17,57),(18,133),(20,155),(21,41),(22,177),(24,139),(26,161),(27,47),(28,123),(30,145),(32,167),(33,53),(34,129),(36,151),(38,173),(39,59),(40,135),(42,157),(44,179),(46,141),(48,163),(50,125),(52,147),(54,169),(56,131),(58,153),(60,175),(61,71),(62,206),(63,93),(64,228),(65,115),(66,190),(67,77),(68,212),(69,99),(70,234),(72,196),(73,83),(74,218),(75,105),(76,240),(78,202),(79,89),(80,224),(81,111),(82,186),(84,208),(85,95),(86,230),(87,117),(88,192),(90,214),(91,101),(92,236),(94,198),(96,220),(97,107),(98,182),(100,204),(102,226),(103,113),(104,188),(106,210),(108,232),(109,119),(110,194),(112,216),(114,238),(116,200),(118,222),(120,184),(122,142),(124,164),(128,148),(130,170),(134,154),(136,176),(140,160),(146,166),(152,172),(158,178),(181,211),(183,233),(185,195),(187,217),(189,239),(191,201),(193,223),(197,207),(199,229),(203,213),(205,235),(209,219),(215,225),(221,231),(227,237)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 4 | 6 | 6 | 10 | 10 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D60⋊C2 | D12⋊5D5 | C30.C23 |
| kernel | C60⋊5C4⋊C2 | Dic3×Dic5 | D6⋊Dic5 | Dic15⋊5C4 | C3×C10.D4 | C5×D6⋊C4 | C60⋊5C4 | C10.D4 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 6 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of C60⋊5C4⋊C2 ►in GL4(𝔽61) generated by
| 8 | 0 | 0 | 0 |
| 52 | 23 | 0 | 0 |
| 0 | 0 | 23 | 38 |
| 0 | 0 | 23 | 46 |
| 3 | 56 | 0 | 0 |
| 26 | 58 | 0 | 0 |
| 0 | 0 | 43 | 52 |
| 0 | 0 | 9 | 18 |
| 1 | 0 | 0 | 0 |
| 50 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [8,52,0,0,0,23,0,0,0,0,23,23,0,0,38,46],[3,26,0,0,56,58,0,0,0,0,43,9,0,0,52,18],[1,50,0,0,0,60,0,0,0,0,0,60,0,0,60,0] >;
C60⋊5C4⋊C2 in GAP, Magma, Sage, TeX
C_{60}\rtimes_5C_4\rtimes C_2 % in TeX
G:=Group("C60:5C4:C2"); // GroupNames label
G:=SmallGroup(480,418);
// by ID
G=gap.SmallGroup(480,418);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,590,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^41*b^2,c*b*c=a^30*b>;
// generators/relations