metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.44D4, C12.19D20, (C4×Dic3)⋊3D5, (C2×D20).4S3, (C6×D20).4C2, C6.56(C2×D20), (Dic3×C20)⋊3C2, (C2×C20).292D6, C30.110(C2×D4), C3⋊4(C4.D20), C15⋊4(C4.4D4), C4.9(C3⋊D20), (C22×D5).9D6, (C2×Dic30)⋊22C2, C6.49(C4○D20), C30.30(C4○D4), D10⋊Dic3⋊5C2, (C2×C12).112D10, C5⋊1(C23.12D6), C20.57(C3⋊D4), (C2×C30).54C23, (C2×C60).111C22, C2.10(D20⋊5S3), C10.22(D4⋊2S3), (C2×Dic3).139D10, (C2×Dic15).56C22, (C10×Dic3).163C22, (D5×C2×C6).7C22, (C2×C4).102(S3×D5), C2.15(C2×C3⋊D20), C10.11(C2×C3⋊D4), C22.141(C2×S3×D5), (C2×C6).66(C22×D5), (C2×C10).66(C22×S3), SmallGroup(480,440)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.44D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a41, cac-1=a-1, cbc-1=a30b-1 >
Subgroups: 796 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C5×Dic3, Dic15, C60, C6×D5, C2×C30, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C23.12D6, C3×D20, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, C4.D20, D10⋊Dic3, Dic3×C20, C6×D20, C2×Dic30, C60.44D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, D20, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C2×D20, C4○D20, C23.12D6, C3⋊D20, C2×S3×D5, C4.D20, D20⋊5S3, C2×C3⋊D20, C60.44D4
►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)
(1 168 198 65)(2 149 199 106)(3 130 200 87)(4 171 201 68)(5 152 202 109)(6 133 203 90)(7 174 204 71)(8 155 205 112)(9 136 206 93)(10 177 207 74)(11 158 208 115)(12 139 209 96)(13 180 210 77)(14 161 211 118)(15 142 212 99)(16 123 213 80)(17 164 214 61)(18 145 215 102)(19 126 216 83)(20 167 217 64)(21 148 218 105)(22 129 219 86)(23 170 220 67)(24 151 221 108)(25 132 222 89)(26 173 223 70)(27 154 224 111)(28 135 225 92)(29 176 226 73)(30 157 227 114)(31 138 228 95)(32 179 229 76)(33 160 230 117)(34 141 231 98)(35 122 232 79)(36 163 233 120)(37 144 234 101)(38 125 235 82)(39 166 236 63)(40 147 237 104)(41 128 238 85)(42 169 239 66)(43 150 240 107)(44 131 181 88)(45 172 182 69)(46 153 183 110)(47 134 184 91)(48 175 185 72)(49 156 186 113)(50 137 187 94)(51 178 188 75)(52 159 189 116)(53 140 190 97)(54 121 191 78)(55 162 192 119)(56 143 193 100)(57 124 194 81)(58 165 195 62)(59 146 196 103)(60 127 197 84)
(1 123 31 153)(2 122 32 152)(3 121 33 151)(4 180 34 150)(5 179 35 149)(6 178 36 148)(7 177 37 147)(8 176 38 146)(9 175 39 145)(10 174 40 144)(11 173 41 143)(12 172 42 142)(13 171 43 141)(14 170 44 140)(15 169 45 139)(16 168 46 138)(17 167 47 137)(18 166 48 136)(19 165 49 135)(20 164 50 134)(21 163 51 133)(22 162 52 132)(23 161 53 131)(24 160 54 130)(25 159 55 129)(26 158 56 128)(27 157 57 127)(28 156 58 126)(29 155 59 125)(30 154 60 124)(61 187 91 217)(62 186 92 216)(63 185 93 215)(64 184 94 214)(65 183 95 213)(66 182 96 212)(67 181 97 211)(68 240 98 210)(69 239 99 209)(70 238 100 208)(71 237 101 207)(72 236 102 206)(73 235 103 205)(74 234 104 204)(75 233 105 203)(76 232 106 202)(77 231 107 201)(78 230 108 200)(79 229 109 199)(80 228 110 198)(81 227 111 197)(82 226 112 196)(83 225 113 195)(84 224 114 194)(85 223 115 193)(86 222 116 192)(87 221 117 191)(88 220 118 190)(89 219 119 189)(90 218 120 188)
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), (1,168,198,65)(2,149,199,106)(3,130,200,87)(4,171,201,68)(5,152,202,109)(6,133,203,90)(7,174,204,71)(8,155,205,112)(9,136,206,93)(10,177,207,74)(11,158,208,115)(12,139,209,96)(13,180,210,77)(14,161,211,118)(15,142,212,99)(16,123,213,80)(17,164,214,61)(18,145,215,102)(19,126,216,83)(20,167,217,64)(21,148,218,105)(22,129,219,86)(23,170,220,67)(24,151,221,108)(25,132,222,89)(26,173,223,70)(27,154,224,111)(28,135,225,92)(29,176,226,73)(30,157,227,114)(31,138,228,95)(32,179,229,76)(33,160,230,117)(34,141,231,98)(35,122,232,79)(36,163,233,120)(37,144,234,101)(38,125,235,82)(39,166,236,63)(40,147,237,104)(41,128,238,85)(42,169,239,66)(43,150,240,107)(44,131,181,88)(45,172,182,69)(46,153,183,110)(47,134,184,91)(48,175,185,72)(49,156,186,113)(50,137,187,94)(51,178,188,75)(52,159,189,116)(53,140,190,97)(54,121,191,78)(55,162,192,119)(56,143,193,100)(57,124,194,81)(58,165,195,62)(59,146,196,103)(60,127,197,84), (1,123,31,153)(2,122,32,152)(3,121,33,151)(4,180,34,150)(5,179,35,149)(6,178,36,148)(7,177,37,147)(8,176,38,146)(9,175,39,145)(10,174,40,144)(11,173,41,143)(12,172,42,142)(13,171,43,141)(14,170,44,140)(15,169,45,139)(16,168,46,138)(17,167,47,137)(18,166,48,136)(19,165,49,135)(20,164,50,134)(21,163,51,133)(22,162,52,132)(23,161,53,131)(24,160,54,130)(25,159,55,129)(26,158,56,128)(27,157,57,127)(28,156,58,126)(29,155,59,125)(30,154,60,124)(61,187,91,217)(62,186,92,216)(63,185,93,215)(64,184,94,214)(65,183,95,213)(66,182,96,212)(67,181,97,211)(68,240,98,210)(69,239,99,209)(70,238,100,208)(71,237,101,207)(72,236,102,206)(73,235,103,205)(74,234,104,204)(75,233,105,203)(76,232,106,202)(77,231,107,201)(78,230,108,200)(79,229,109,199)(80,228,110,198)(81,227,111,197)(82,226,112,196)(83,225,113,195)(84,224,114,194)(85,223,115,193)(86,222,116,192)(87,221,117,191)(88,220,118,190)(89,219,119,189)(90,218,120,188)>;
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), (1,168,198,65)(2,149,199,106)(3,130,200,87)(4,171,201,68)(5,152,202,109)(6,133,203,90)(7,174,204,71)(8,155,205,112)(9,136,206,93)(10,177,207,74)(11,158,208,115)(12,139,209,96)(13,180,210,77)(14,161,211,118)(15,142,212,99)(16,123,213,80)(17,164,214,61)(18,145,215,102)(19,126,216,83)(20,167,217,64)(21,148,218,105)(22,129,219,86)(23,170,220,67)(24,151,221,108)(25,132,222,89)(26,173,223,70)(27,154,224,111)(28,135,225,92)(29,176,226,73)(30,157,227,114)(31,138,228,95)(32,179,229,76)(33,160,230,117)(34,141,231,98)(35,122,232,79)(36,163,233,120)(37,144,234,101)(38,125,235,82)(39,166,236,63)(40,147,237,104)(41,128,238,85)(42,169,239,66)(43,150,240,107)(44,131,181,88)(45,172,182,69)(46,153,183,110)(47,134,184,91)(48,175,185,72)(49,156,186,113)(50,137,187,94)(51,178,188,75)(52,159,189,116)(53,140,190,97)(54,121,191,78)(55,162,192,119)(56,143,193,100)(57,124,194,81)(58,165,195,62)(59,146,196,103)(60,127,197,84), (1,123,31,153)(2,122,32,152)(3,121,33,151)(4,180,34,150)(5,179,35,149)(6,178,36,148)(7,177,37,147)(8,176,38,146)(9,175,39,145)(10,174,40,144)(11,173,41,143)(12,172,42,142)(13,171,43,141)(14,170,44,140)(15,169,45,139)(16,168,46,138)(17,167,47,137)(18,166,48,136)(19,165,49,135)(20,164,50,134)(21,163,51,133)(22,162,52,132)(23,161,53,131)(24,160,54,130)(25,159,55,129)(26,158,56,128)(27,157,57,127)(28,156,58,126)(29,155,59,125)(30,154,60,124)(61,187,91,217)(62,186,92,216)(63,185,93,215)(64,184,94,214)(65,183,95,213)(66,182,96,212)(67,181,97,211)(68,240,98,210)(69,239,99,209)(70,238,100,208)(71,237,101,207)(72,236,102,206)(73,235,103,205)(74,234,104,204)(75,233,105,203)(76,232,106,202)(77,231,107,201)(78,230,108,200)(79,229,109,199)(80,228,110,198)(81,227,111,197)(82,226,112,196)(83,225,113,195)(84,224,114,194)(85,223,115,193)(86,222,116,192)(87,221,117,191)(88,220,118,190)(89,219,119,189)(90,218,120,188) );
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)], [(1,168,198,65),(2,149,199,106),(3,130,200,87),(4,171,201,68),(5,152,202,109),(6,133,203,90),(7,174,204,71),(8,155,205,112),(9,136,206,93),(10,177,207,74),(11,158,208,115),(12,139,209,96),(13,180,210,77),(14,161,211,118),(15,142,212,99),(16,123,213,80),(17,164,214,61),(18,145,215,102),(19,126,216,83),(20,167,217,64),(21,148,218,105),(22,129,219,86),(23,170,220,67),(24,151,221,108),(25,132,222,89),(26,173,223,70),(27,154,224,111),(28,135,225,92),(29,176,226,73),(30,157,227,114),(31,138,228,95),(32,179,229,76),(33,160,230,117),(34,141,231,98),(35,122,232,79),(36,163,233,120),(37,144,234,101),(38,125,235,82),(39,166,236,63),(40,147,237,104),(41,128,238,85),(42,169,239,66),(43,150,240,107),(44,131,181,88),(45,172,182,69),(46,153,183,110),(47,134,184,91),(48,175,185,72),(49,156,186,113),(50,137,187,94),(51,178,188,75),(52,159,189,116),(53,140,190,97),(54,121,191,78),(55,162,192,119),(56,143,193,100),(57,124,194,81),(58,165,195,62),(59,146,196,103),(60,127,197,84)], [(1,123,31,153),(2,122,32,152),(3,121,33,151),(4,180,34,150),(5,179,35,149),(6,178,36,148),(7,177,37,147),(8,176,38,146),(9,175,39,145),(10,174,40,144),(11,173,41,143),(12,172,42,142),(13,171,43,141),(14,170,44,140),(15,169,45,139),(16,168,46,138),(17,167,47,137),(18,166,48,136),(19,165,49,135),(20,164,50,134),(21,163,51,133),(22,162,52,132),(23,161,53,131),(24,160,54,130),(25,159,55,129),(26,158,56,128),(27,157,57,127),(28,156,58,126),(29,155,59,125),(30,154,60,124),(61,187,91,217),(62,186,92,216),(63,185,93,215),(64,184,94,214),(65,183,95,213),(66,182,96,212),(67,181,97,211),(68,240,98,210),(69,239,99,209),(70,238,100,208),(71,237,101,207),(72,236,102,206),(73,235,103,205),(74,234,104,204),(75,233,105,203),(76,232,106,202),(77,231,107,201),(78,230,108,200),(79,229,109,199),(80,228,110,198),(81,227,111,197),(82,226,112,196),(83,225,113,195),(84,224,114,194),(85,223,115,193),(86,222,116,192),(87,221,117,191),(88,220,118,190),(89,219,119,189),(90,218,120,188)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D20 | C4○D20 | D4⋊2S3 | S3×D5 | C3⋊D20 | C2×S3×D5 | D20⋊5S3 |
| kernel | C60.44D4 | D10⋊Dic3 | Dic3×C20 | C6×D20 | C2×Dic30 | C2×D20 | C60 | C4×Dic3 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C60.44D4 ►in GL4(𝔽61) generated by
| 48 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 57 | 2 |
| 0 | 0 | 23 | 34 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 30 | 45 |
| 0 | 0 | 60 | 31 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 11 |
| 0 | 0 | 46 | 15 |
G:=sub<GL(4,GF(61))| [48,0,0,0,0,14,0,0,0,0,57,23,0,0,2,34],[0,60,0,0,1,0,0,0,0,0,30,60,0,0,45,31],[0,1,0,0,1,0,0,0,0,0,46,46,0,0,11,15] >;
C60.44D4 in GAP, Magma, Sage, TeX
C_{60}._{44}D_4 % in TeX
G:=Group("C60.44D4"); // GroupNames label
G:=SmallGroup(480,440);
// by ID
G=gap.SmallGroup(480,440);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^41,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations