metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.77D4, C30.15D8, C20.7D12, D12⋊3Dic5, C30.3SD16, (C5×D12)⋊10C4, C20.33(C4×S3), C4⋊Dic5⋊10S3, (C2×D12).6D5, (C2×C30).13D4, (C2×C20).47D6, C5⋊4(C6.D8), C4.6(S3×Dic5), C15⋊5(D4⋊C4), C60.121(C2×C4), C6.13(D4⋊D5), (C10×D12).6C2, (C2×C12).49D10, C2.2(C15⋊D8), C3⋊2(D4⋊Dic5), C6.4(D4.D5), C12.1(C2×Dic5), C10.41(D6⋊C4), C10.13(D4⋊S3), C4.21(C5⋊D12), C12.12(C5⋊D4), C2.5(D6⋊Dic5), C6.4(C23.D5), C30.49(C22⋊C4), (C2×C60).182C22, C10.4(Q8⋊2S3), C2.1(C30.D4), C22.13(C15⋊D4), (C3×C4⋊Dic5)⋊7C2, (C2×C15⋊3C8)⋊14C2, (C2×C4).185(S3×D5), (C2×C6).44(C5⋊D4), (C2×C10).44(C3⋊D4), SmallGroup(480,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.77D4
G = < a,b,c | a60=b4=1, c2=a15, bab-1=a19, cac-1=a29, cbc-1=a15b-1 >
Subgroups: 428 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, D12 [×2], D12, C2×C12, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], D4⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60 [×2], S3×C10 [×4], C2×C30, C2×C5⋊2C8, C4⋊Dic5, D4×C10, C6.D8, C15⋊3C8, C6×Dic5, C5×D12 [×2], C5×D12, C2×C60, S3×C2×C10, D4⋊Dic5, C3×C4⋊Dic5, C2×C15⋊3C8, C10×D12, C60.77D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, D4⋊C4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, D4⋊S3, Q8⋊2S3, S3×D5, D4⋊D5, D4.D5, C23.D5, C6.D8, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊Dic5, C15⋊D8, C30.D4, D6⋊Dic5, C60.77D4
►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)
(1 146 240 90)(2 165 181 109)(3 124 182 68)(4 143 183 87)(5 162 184 106)(6 121 185 65)(7 140 186 84)(8 159 187 103)(9 178 188 62)(10 137 189 81)(11 156 190 100)(12 175 191 119)(13 134 192 78)(14 153 193 97)(15 172 194 116)(16 131 195 75)(17 150 196 94)(18 169 197 113)(19 128 198 72)(20 147 199 91)(21 166 200 110)(22 125 201 69)(23 144 202 88)(24 163 203 107)(25 122 204 66)(26 141 205 85)(27 160 206 104)(28 179 207 63)(29 138 208 82)(30 157 209 101)(31 176 210 120)(32 135 211 79)(33 154 212 98)(34 173 213 117)(35 132 214 76)(36 151 215 95)(37 170 216 114)(38 129 217 73)(39 148 218 92)(40 167 219 111)(41 126 220 70)(42 145 221 89)(43 164 222 108)(44 123 223 67)(45 142 224 86)(46 161 225 105)(47 180 226 64)(48 139 227 83)(49 158 228 102)(50 177 229 61)(51 136 230 80)(52 155 231 99)(53 174 232 118)(54 133 233 77)(55 152 234 96)(56 171 235 115)(57 130 236 74)(58 149 237 93)(59 168 238 112)(60 127 239 71)
(1 161 16 176 31 131 46 146)(2 130 17 145 32 160 47 175)(3 159 18 174 33 129 48 144)(4 128 19 143 34 158 49 173)(5 157 20 172 35 127 50 142)(6 126 21 141 36 156 51 171)(7 155 22 170 37 125 52 140)(8 124 23 139 38 154 53 169)(9 153 24 168 39 123 54 138)(10 122 25 137 40 152 55 167)(11 151 26 166 41 121 56 136)(12 180 27 135 42 150 57 165)(13 149 28 164 43 179 58 134)(14 178 29 133 44 148 59 163)(15 147 30 162 45 177 60 132)(61 239 76 194 91 209 106 224)(62 208 77 223 92 238 107 193)(63 237 78 192 93 207 108 222)(64 206 79 221 94 236 109 191)(65 235 80 190 95 205 110 220)(66 204 81 219 96 234 111 189)(67 233 82 188 97 203 112 218)(68 202 83 217 98 232 113 187)(69 231 84 186 99 201 114 216)(70 200 85 215 100 230 115 185)(71 229 86 184 101 199 116 214)(72 198 87 213 102 228 117 183)(73 227 88 182 103 197 118 212)(74 196 89 211 104 226 119 181)(75 225 90 240 105 195 120 210)
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), (1,146,240,90)(2,165,181,109)(3,124,182,68)(4,143,183,87)(5,162,184,106)(6,121,185,65)(7,140,186,84)(8,159,187,103)(9,178,188,62)(10,137,189,81)(11,156,190,100)(12,175,191,119)(13,134,192,78)(14,153,193,97)(15,172,194,116)(16,131,195,75)(17,150,196,94)(18,169,197,113)(19,128,198,72)(20,147,199,91)(21,166,200,110)(22,125,201,69)(23,144,202,88)(24,163,203,107)(25,122,204,66)(26,141,205,85)(27,160,206,104)(28,179,207,63)(29,138,208,82)(30,157,209,101)(31,176,210,120)(32,135,211,79)(33,154,212,98)(34,173,213,117)(35,132,214,76)(36,151,215,95)(37,170,216,114)(38,129,217,73)(39,148,218,92)(40,167,219,111)(41,126,220,70)(42,145,221,89)(43,164,222,108)(44,123,223,67)(45,142,224,86)(46,161,225,105)(47,180,226,64)(48,139,227,83)(49,158,228,102)(50,177,229,61)(51,136,230,80)(52,155,231,99)(53,174,232,118)(54,133,233,77)(55,152,234,96)(56,171,235,115)(57,130,236,74)(58,149,237,93)(59,168,238,112)(60,127,239,71), (1,161,16,176,31,131,46,146)(2,130,17,145,32,160,47,175)(3,159,18,174,33,129,48,144)(4,128,19,143,34,158,49,173)(5,157,20,172,35,127,50,142)(6,126,21,141,36,156,51,171)(7,155,22,170,37,125,52,140)(8,124,23,139,38,154,53,169)(9,153,24,168,39,123,54,138)(10,122,25,137,40,152,55,167)(11,151,26,166,41,121,56,136)(12,180,27,135,42,150,57,165)(13,149,28,164,43,179,58,134)(14,178,29,133,44,148,59,163)(15,147,30,162,45,177,60,132)(61,239,76,194,91,209,106,224)(62,208,77,223,92,238,107,193)(63,237,78,192,93,207,108,222)(64,206,79,221,94,236,109,191)(65,235,80,190,95,205,110,220)(66,204,81,219,96,234,111,189)(67,233,82,188,97,203,112,218)(68,202,83,217,98,232,113,187)(69,231,84,186,99,201,114,216)(70,200,85,215,100,230,115,185)(71,229,86,184,101,199,116,214)(72,198,87,213,102,228,117,183)(73,227,88,182,103,197,118,212)(74,196,89,211,104,226,119,181)(75,225,90,240,105,195,120,210)>;
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), (1,146,240,90)(2,165,181,109)(3,124,182,68)(4,143,183,87)(5,162,184,106)(6,121,185,65)(7,140,186,84)(8,159,187,103)(9,178,188,62)(10,137,189,81)(11,156,190,100)(12,175,191,119)(13,134,192,78)(14,153,193,97)(15,172,194,116)(16,131,195,75)(17,150,196,94)(18,169,197,113)(19,128,198,72)(20,147,199,91)(21,166,200,110)(22,125,201,69)(23,144,202,88)(24,163,203,107)(25,122,204,66)(26,141,205,85)(27,160,206,104)(28,179,207,63)(29,138,208,82)(30,157,209,101)(31,176,210,120)(32,135,211,79)(33,154,212,98)(34,173,213,117)(35,132,214,76)(36,151,215,95)(37,170,216,114)(38,129,217,73)(39,148,218,92)(40,167,219,111)(41,126,220,70)(42,145,221,89)(43,164,222,108)(44,123,223,67)(45,142,224,86)(46,161,225,105)(47,180,226,64)(48,139,227,83)(49,158,228,102)(50,177,229,61)(51,136,230,80)(52,155,231,99)(53,174,232,118)(54,133,233,77)(55,152,234,96)(56,171,235,115)(57,130,236,74)(58,149,237,93)(59,168,238,112)(60,127,239,71), (1,161,16,176,31,131,46,146)(2,130,17,145,32,160,47,175)(3,159,18,174,33,129,48,144)(4,128,19,143,34,158,49,173)(5,157,20,172,35,127,50,142)(6,126,21,141,36,156,51,171)(7,155,22,170,37,125,52,140)(8,124,23,139,38,154,53,169)(9,153,24,168,39,123,54,138)(10,122,25,137,40,152,55,167)(11,151,26,166,41,121,56,136)(12,180,27,135,42,150,57,165)(13,149,28,164,43,179,58,134)(14,178,29,133,44,148,59,163)(15,147,30,162,45,177,60,132)(61,239,76,194,91,209,106,224)(62,208,77,223,92,238,107,193)(63,237,78,192,93,207,108,222)(64,206,79,221,94,236,109,191)(65,235,80,190,95,205,110,220)(66,204,81,219,96,234,111,189)(67,233,82,188,97,203,112,218)(68,202,83,217,98,232,113,187)(69,231,84,186,99,201,114,216)(70,200,85,215,100,230,115,185)(71,229,86,184,101,199,116,214)(72,198,87,213,102,228,117,183)(73,227,88,182,103,197,118,212)(74,196,89,211,104,226,119,181)(75,225,90,240,105,195,120,210) );
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)], [(1,146,240,90),(2,165,181,109),(3,124,182,68),(4,143,183,87),(5,162,184,106),(6,121,185,65),(7,140,186,84),(8,159,187,103),(9,178,188,62),(10,137,189,81),(11,156,190,100),(12,175,191,119),(13,134,192,78),(14,153,193,97),(15,172,194,116),(16,131,195,75),(17,150,196,94),(18,169,197,113),(19,128,198,72),(20,147,199,91),(21,166,200,110),(22,125,201,69),(23,144,202,88),(24,163,203,107),(25,122,204,66),(26,141,205,85),(27,160,206,104),(28,179,207,63),(29,138,208,82),(30,157,209,101),(31,176,210,120),(32,135,211,79),(33,154,212,98),(34,173,213,117),(35,132,214,76),(36,151,215,95),(37,170,216,114),(38,129,217,73),(39,148,218,92),(40,167,219,111),(41,126,220,70),(42,145,221,89),(43,164,222,108),(44,123,223,67),(45,142,224,86),(46,161,225,105),(47,180,226,64),(48,139,227,83),(49,158,228,102),(50,177,229,61),(51,136,230,80),(52,155,231,99),(53,174,232,118),(54,133,233,77),(55,152,234,96),(56,171,235,115),(57,130,236,74),(58,149,237,93),(59,168,238,112),(60,127,239,71)], [(1,161,16,176,31,131,46,146),(2,130,17,145,32,160,47,175),(3,159,18,174,33,129,48,144),(4,128,19,143,34,158,49,173),(5,157,20,172,35,127,50,142),(6,126,21,141,36,156,51,171),(7,155,22,170,37,125,52,140),(8,124,23,139,38,154,53,169),(9,153,24,168,39,123,54,138),(10,122,25,137,40,152,55,167),(11,151,26,166,41,121,56,136),(12,180,27,135,42,150,57,165),(13,149,28,164,43,179,58,134),(14,178,29,133,44,148,59,163),(15,147,30,162,45,177,60,132),(61,239,76,194,91,209,106,224),(62,208,77,223,92,238,107,193),(63,237,78,192,93,207,108,222),(64,206,79,221,94,236,109,191),(65,235,80,190,95,205,110,220),(66,204,81,219,96,234,111,189),(67,233,82,188,97,203,112,218),(68,202,83,217,98,232,113,187),(69,231,84,186,99,201,114,216),(70,200,85,215,100,230,115,185),(71,229,86,184,101,199,116,214),(72,198,87,213,102,228,117,183),(73,227,88,182,103,197,118,212),(74,196,89,211,104,226,119,181),(75,225,90,240,105,195,120,210)])
60 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 | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 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 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | Dic5 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | C5⋊D4 | D4⋊S3 | Q8⋊2S3 | S3×D5 | D4⋊D5 | D4.D5 | S3×Dic5 | C5⋊D12 | C15⋊D4 | C15⋊D8 | C30.D4 |
| kernel | C60.77D4 | C3×C4⋊Dic5 | C2×C15⋊3C8 | C10×D12 | C5×D12 | C4⋊Dic5 | C60 | C2×C30 | C2×D12 | C2×C20 | C30 | C30 | D12 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.77D4 ►in GL6(𝔽241)
| 2 | 49 | 0 | 0 | 0 | 0 |
| 177 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 51 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 164 |
| 0 | 0 | 0 | 0 | 144 | 1 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 166 | 0 | 0 |
| 0 | 0 | 104 | 150 | 0 | 0 |
| 0 | 0 | 0 | 0 | 203 | 224 |
| 0 | 0 | 0 | 0 | 85 | 38 |
| 177 | 238 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 150 | 75 | 0 | 0 |
| 0 | 0 | 137 | 91 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 224 |
| 0 | 0 | 0 | 0 | 85 | 38 |
G:=sub<GL(6,GF(241))| [2,177,0,0,0,0,49,240,0,0,0,0,0,0,1,190,0,0,0,0,51,51,0,0,0,0,0,0,240,144,0,0,0,0,164,1],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,91,104,0,0,0,0,166,150,0,0,0,0,0,0,203,85,0,0,0,0,224,38],[177,0,0,0,0,0,238,64,0,0,0,0,0,0,150,137,0,0,0,0,75,91,0,0,0,0,0,0,0,85,0,0,0,0,224,38] >;
C60.77D4 in GAP, Magma, Sage, TeX
C_{60}._{77}D_4 % in TeX
G:=Group("C60.77D4"); // GroupNames label
G:=SmallGroup(480,42);
// by ID
G=gap.SmallGroup(480,42);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^15,b*a*b^-1=a^19,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;
// generators/relations