metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.45D4, D6⋊6Dic10, C20.53D12, C4⋊Dic5⋊4S3, (S3×C10)⋊6Q8, C5⋊5(C4.D12), C15⋊5(C22⋊Q8), C30.23(C2×Q8), C10.27(S3×Q8), (C2×C20).293D6, C10.56(C2×D12), C30.111(C2×D4), D6⋊Dic5.5C2, C30.Q8⋊6C2, C6.9(C2×Dic10), (C2×Dic30)⋊23C2, C30.31(C4○D4), C6.50(C4○D20), (C2×C12).113D10, C12.35(C5⋊D4), C3⋊2(C20.48D4), C4.18(C5⋊D12), (C2×C30).55C23, (C2×Dic5).13D6, C2.11(S3×Dic10), (C2×C60).112C22, (C22×S3).66D10, C10.23(D4⋊2S3), C2.11(D20⋊5S3), (C2×Dic3).140D10, (C6×Dic5).32C22, (C2×Dic15).57C22, (C10×Dic3).164C22, (S3×C2×C4).3D5, (S3×C2×C20).3C2, (C3×C4⋊Dic5)⋊3C2, C6.10(C2×C5⋊D4), (C2×C4).103(S3×D5), C2.14(C2×C5⋊D12), C22.142(C2×S3×D5), (S3×C2×C10).78C22, (C2×C6).67(C22×D5), (C2×C10).67(C22×S3), SmallGroup(480,441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.45D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a19, cac-1=a-1, cbc-1=a30b-1 >
Subgroups: 652 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C4.D12, C6×Dic5, S3×C20, C10×Dic3, Dic30, C2×Dic15, C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5, C30.Q8, C3×C4⋊Dic5, S3×C2×C20, C2×Dic30, C60.45D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, D12, C22×S3, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C2×D12, D4⋊2S3, S3×Q8, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C4.D12, C5⋊D12, C2×S3×D5, C20.48D4, D20⋊5S3, S3×Dic10, C2×C5⋊D12, C60.45D4
►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)
(1 125 228 91)(2 144 229 110)(3 163 230 69)(4 122 231 88)(5 141 232 107)(6 160 233 66)(7 179 234 85)(8 138 235 104)(9 157 236 63)(10 176 237 82)(11 135 238 101)(12 154 239 120)(13 173 240 79)(14 132 181 98)(15 151 182 117)(16 170 183 76)(17 129 184 95)(18 148 185 114)(19 167 186 73)(20 126 187 92)(21 145 188 111)(22 164 189 70)(23 123 190 89)(24 142 191 108)(25 161 192 67)(26 180 193 86)(27 139 194 105)(28 158 195 64)(29 177 196 83)(30 136 197 102)(31 155 198 61)(32 174 199 80)(33 133 200 99)(34 152 201 118)(35 171 202 77)(36 130 203 96)(37 149 204 115)(38 168 205 74)(39 127 206 93)(40 146 207 112)(41 165 208 71)(42 124 209 90)(43 143 210 109)(44 162 211 68)(45 121 212 87)(46 140 213 106)(47 159 214 65)(48 178 215 84)(49 137 216 103)(50 156 217 62)(51 175 218 81)(52 134 219 100)(53 153 220 119)(54 172 221 78)(55 131 222 97)(56 150 223 116)(57 169 224 75)(58 128 225 94)(59 147 226 113)(60 166 227 72)
(1 61 31 91)(2 120 32 90)(3 119 33 89)(4 118 34 88)(5 117 35 87)(6 116 36 86)(7 115 37 85)(8 114 38 84)(9 113 39 83)(10 112 40 82)(11 111 41 81)(12 110 42 80)(13 109 43 79)(14 108 44 78)(15 107 45 77)(16 106 46 76)(17 105 47 75)(18 104 48 74)(19 103 49 73)(20 102 50 72)(21 101 51 71)(22 100 52 70)(23 99 53 69)(24 98 54 68)(25 97 55 67)(26 96 56 66)(27 95 57 65)(28 94 58 64)(29 93 59 63)(30 92 60 62)(121 232 151 202)(122 231 152 201)(123 230 153 200)(124 229 154 199)(125 228 155 198)(126 227 156 197)(127 226 157 196)(128 225 158 195)(129 224 159 194)(130 223 160 193)(131 222 161 192)(132 221 162 191)(133 220 163 190)(134 219 164 189)(135 218 165 188)(136 217 166 187)(137 216 167 186)(138 215 168 185)(139 214 169 184)(140 213 170 183)(141 212 171 182)(142 211 172 181)(143 210 173 240)(144 209 174 239)(145 208 175 238)(146 207 176 237)(147 206 177 236)(148 205 178 235)(149 204 179 234)(150 203 180 233)
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), (1,125,228,91)(2,144,229,110)(3,163,230,69)(4,122,231,88)(5,141,232,107)(6,160,233,66)(7,179,234,85)(8,138,235,104)(9,157,236,63)(10,176,237,82)(11,135,238,101)(12,154,239,120)(13,173,240,79)(14,132,181,98)(15,151,182,117)(16,170,183,76)(17,129,184,95)(18,148,185,114)(19,167,186,73)(20,126,187,92)(21,145,188,111)(22,164,189,70)(23,123,190,89)(24,142,191,108)(25,161,192,67)(26,180,193,86)(27,139,194,105)(28,158,195,64)(29,177,196,83)(30,136,197,102)(31,155,198,61)(32,174,199,80)(33,133,200,99)(34,152,201,118)(35,171,202,77)(36,130,203,96)(37,149,204,115)(38,168,205,74)(39,127,206,93)(40,146,207,112)(41,165,208,71)(42,124,209,90)(43,143,210,109)(44,162,211,68)(45,121,212,87)(46,140,213,106)(47,159,214,65)(48,178,215,84)(49,137,216,103)(50,156,217,62)(51,175,218,81)(52,134,219,100)(53,153,220,119)(54,172,221,78)(55,131,222,97)(56,150,223,116)(57,169,224,75)(58,128,225,94)(59,147,226,113)(60,166,227,72), (1,61,31,91)(2,120,32,90)(3,119,33,89)(4,118,34,88)(5,117,35,87)(6,116,36,86)(7,115,37,85)(8,114,38,84)(9,113,39,83)(10,112,40,82)(11,111,41,81)(12,110,42,80)(13,109,43,79)(14,108,44,78)(15,107,45,77)(16,106,46,76)(17,105,47,75)(18,104,48,74)(19,103,49,73)(20,102,50,72)(21,101,51,71)(22,100,52,70)(23,99,53,69)(24,98,54,68)(25,97,55,67)(26,96,56,66)(27,95,57,65)(28,94,58,64)(29,93,59,63)(30,92,60,62)(121,232,151,202)(122,231,152,201)(123,230,153,200)(124,229,154,199)(125,228,155,198)(126,227,156,197)(127,226,157,196)(128,225,158,195)(129,224,159,194)(130,223,160,193)(131,222,161,192)(132,221,162,191)(133,220,163,190)(134,219,164,189)(135,218,165,188)(136,217,166,187)(137,216,167,186)(138,215,168,185)(139,214,169,184)(140,213,170,183)(141,212,171,182)(142,211,172,181)(143,210,173,240)(144,209,174,239)(145,208,175,238)(146,207,176,237)(147,206,177,236)(148,205,178,235)(149,204,179,234)(150,203,180,233)>;
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), (1,125,228,91)(2,144,229,110)(3,163,230,69)(4,122,231,88)(5,141,232,107)(6,160,233,66)(7,179,234,85)(8,138,235,104)(9,157,236,63)(10,176,237,82)(11,135,238,101)(12,154,239,120)(13,173,240,79)(14,132,181,98)(15,151,182,117)(16,170,183,76)(17,129,184,95)(18,148,185,114)(19,167,186,73)(20,126,187,92)(21,145,188,111)(22,164,189,70)(23,123,190,89)(24,142,191,108)(25,161,192,67)(26,180,193,86)(27,139,194,105)(28,158,195,64)(29,177,196,83)(30,136,197,102)(31,155,198,61)(32,174,199,80)(33,133,200,99)(34,152,201,118)(35,171,202,77)(36,130,203,96)(37,149,204,115)(38,168,205,74)(39,127,206,93)(40,146,207,112)(41,165,208,71)(42,124,209,90)(43,143,210,109)(44,162,211,68)(45,121,212,87)(46,140,213,106)(47,159,214,65)(48,178,215,84)(49,137,216,103)(50,156,217,62)(51,175,218,81)(52,134,219,100)(53,153,220,119)(54,172,221,78)(55,131,222,97)(56,150,223,116)(57,169,224,75)(58,128,225,94)(59,147,226,113)(60,166,227,72), (1,61,31,91)(2,120,32,90)(3,119,33,89)(4,118,34,88)(5,117,35,87)(6,116,36,86)(7,115,37,85)(8,114,38,84)(9,113,39,83)(10,112,40,82)(11,111,41,81)(12,110,42,80)(13,109,43,79)(14,108,44,78)(15,107,45,77)(16,106,46,76)(17,105,47,75)(18,104,48,74)(19,103,49,73)(20,102,50,72)(21,101,51,71)(22,100,52,70)(23,99,53,69)(24,98,54,68)(25,97,55,67)(26,96,56,66)(27,95,57,65)(28,94,58,64)(29,93,59,63)(30,92,60,62)(121,232,151,202)(122,231,152,201)(123,230,153,200)(124,229,154,199)(125,228,155,198)(126,227,156,197)(127,226,157,196)(128,225,158,195)(129,224,159,194)(130,223,160,193)(131,222,161,192)(132,221,162,191)(133,220,163,190)(134,219,164,189)(135,218,165,188)(136,217,166,187)(137,216,167,186)(138,215,168,185)(139,214,169,184)(140,213,170,183)(141,212,171,182)(142,211,172,181)(143,210,173,240)(144,209,174,239)(145,208,175,238)(146,207,176,237)(147,206,177,236)(148,205,178,235)(149,204,179,234)(150,203,180,233) );
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)], [(1,125,228,91),(2,144,229,110),(3,163,230,69),(4,122,231,88),(5,141,232,107),(6,160,233,66),(7,179,234,85),(8,138,235,104),(9,157,236,63),(10,176,237,82),(11,135,238,101),(12,154,239,120),(13,173,240,79),(14,132,181,98),(15,151,182,117),(16,170,183,76),(17,129,184,95),(18,148,185,114),(19,167,186,73),(20,126,187,92),(21,145,188,111),(22,164,189,70),(23,123,190,89),(24,142,191,108),(25,161,192,67),(26,180,193,86),(27,139,194,105),(28,158,195,64),(29,177,196,83),(30,136,197,102),(31,155,198,61),(32,174,199,80),(33,133,200,99),(34,152,201,118),(35,171,202,77),(36,130,203,96),(37,149,204,115),(38,168,205,74),(39,127,206,93),(40,146,207,112),(41,165,208,71),(42,124,209,90),(43,143,210,109),(44,162,211,68),(45,121,212,87),(46,140,213,106),(47,159,214,65),(48,178,215,84),(49,137,216,103),(50,156,217,62),(51,175,218,81),(52,134,219,100),(53,153,220,119),(54,172,221,78),(55,131,222,97),(56,150,223,116),(57,169,224,75),(58,128,225,94),(59,147,226,113),(60,166,227,72)], [(1,61,31,91),(2,120,32,90),(3,119,33,89),(4,118,34,88),(5,117,35,87),(6,116,36,86),(7,115,37,85),(8,114,38,84),(9,113,39,83),(10,112,40,82),(11,111,41,81),(12,110,42,80),(13,109,43,79),(14,108,44,78),(15,107,45,77),(16,106,46,76),(17,105,47,75),(18,104,48,74),(19,103,49,73),(20,102,50,72),(21,101,51,71),(22,100,52,70),(23,99,53,69),(24,98,54,68),(25,97,55,67),(26,96,56,66),(27,95,57,65),(28,94,58,64),(29,93,59,63),(30,92,60,62),(121,232,151,202),(122,231,152,201),(123,230,153,200),(124,229,154,199),(125,228,155,198),(126,227,156,197),(127,226,157,196),(128,225,158,195),(129,224,159,194),(130,223,160,193),(131,222,161,192),(132,221,162,191),(133,220,163,190),(134,219,164,189),(135,218,165,188),(136,217,166,187),(137,216,167,186),(138,215,168,185),(139,214,169,184),(140,213,170,183),(141,212,171,182),(142,211,172,181),(143,210,173,240),(144,209,174,239),(145,208,175,238),(146,207,176,237),(147,206,177,236),(148,205,178,235),(149,204,179,234),(150,203,180,233)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 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 | 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 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 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 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | - | - | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D12 | C5⋊D4 | Dic10 | C4○D20 | D4⋊2S3 | S3×Q8 | S3×D5 | C5⋊D12 | C2×S3×D5 | D20⋊5S3 | S3×Dic10 |
| kernel | C60.45D4 | D6⋊Dic5 | C30.Q8 | C3×C4⋊Dic5 | S3×C2×C20 | C2×Dic30 | C4⋊Dic5 | C60 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C12 | D6 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60.45D4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 11 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 54 | 0 | 0 |
| 0 | 0 | 36 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 3 |
| 0 | 0 | 0 | 0 | 45 | 13 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 54 | 0 | 0 |
| 0 | 0 | 45 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 3 |
| 0 | 0 | 0 | 0 | 5 | 13 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,8,11,0,0,0,0,0,23,0,0,0,0,0,0,52,27,0,0,0,0,0,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,3,36,0,0,0,0,54,58,0,0,0,0,0,0,48,45,0,0,0,0,3,13],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,3,45,0,0,0,0,54,58,0,0,0,0,0,0,48,5,0,0,0,0,3,13] >;
C60.45D4 in GAP, Magma, Sage, TeX
C_{60}._{45}D_4 % in TeX
G:=Group("C60.45D4"); // GroupNames label
G:=SmallGroup(480,441);
// by ID
G=gap.SmallGroup(480,441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,590,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations