metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.89D4, D6⋊Dic5⋊5C2, (C2×D12).8D5, (C10×D12).8C2, (C2×C20).114D6, C30.114(C2×D4), C15⋊6(C4.4D4), (C2×Dic10)⋊10S3, (C6×Dic10)⋊10C2, (C4×Dic15)⋊24C2, C30.36(C4○D4), (C2×C12).116D10, C6.8(D4⋊2D5), C5⋊3(C12.23D4), C20.36(C3⋊D4), C3⋊3(C20.17D4), C4.10(C15⋊D4), C12.38(C5⋊D4), (C2×C30).60C23, (C2×Dic5).17D6, (C22×S3).7D10, (C2×C60).195C22, C2.12(D12⋊D5), C10.27(Q8⋊3S3), (C6×Dic5).36C22, (C2×Dic15).189C22, C6.83(C2×C5⋊D4), (C2×C4).205(S3×D5), C10.84(C2×C3⋊D4), C2.17(C2×C15⋊D4), (S3×C2×C10).7C22, C22.147(C2×S3×D5), (C2×C6).72(C22×D5), (C2×C10).72(C22×S3), SmallGroup(480,446)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.89D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a29, cac-1=a19, cbc-1=a30b-1 >
Subgroups: 668 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C23.D5, C2×Dic10, D4×C10, C12.23D4, C3×Dic10, C6×Dic5, C5×D12, C2×Dic15, C2×C60, S3×C2×C10, C20.17D4, D6⋊Dic5, C4×Dic15, C6×Dic10, C10×D12, C60.89D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, C5⋊D4, C22×D5, Q8⋊3S3, C2×C3⋊D4, S3×D5, D4⋊2D5, C2×C5⋊D4, C12.23D4, C15⋊D4, C2×S3×D5, C20.17D4, D12⋊D5, C2×C15⋊D4, C60.89D4
►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)
(1 160 236 98)(2 129 237 67)(3 158 238 96)(4 127 239 65)(5 156 240 94)(6 125 181 63)(7 154 182 92)(8 123 183 61)(9 152 184 90)(10 121 185 119)(11 150 186 88)(12 179 187 117)(13 148 188 86)(14 177 189 115)(15 146 190 84)(16 175 191 113)(17 144 192 82)(18 173 193 111)(19 142 194 80)(20 171 195 109)(21 140 196 78)(22 169 197 107)(23 138 198 76)(24 167 199 105)(25 136 200 74)(26 165 201 103)(27 134 202 72)(28 163 203 101)(29 132 204 70)(30 161 205 99)(31 130 206 68)(32 159 207 97)(33 128 208 66)(34 157 209 95)(35 126 210 64)(36 155 211 93)(37 124 212 62)(38 153 213 91)(39 122 214 120)(40 151 215 89)(41 180 216 118)(42 149 217 87)(43 178 218 116)(44 147 219 85)(45 176 220 114)(46 145 221 83)(47 174 222 112)(48 143 223 81)(49 172 224 110)(50 141 225 79)(51 170 226 108)(52 139 227 77)(53 168 228 106)(54 137 229 75)(55 166 230 104)(56 135 231 73)(57 164 232 102)(58 133 233 71)(59 162 234 100)(60 131 235 69)
(1 175 31 145)(2 134 32 164)(3 153 33 123)(4 172 34 142)(5 131 35 161)(6 150 36 180)(7 169 37 139)(8 128 38 158)(9 147 39 177)(10 166 40 136)(11 125 41 155)(12 144 42 174)(13 163 43 133)(14 122 44 152)(15 141 45 171)(16 160 46 130)(17 179 47 149)(18 138 48 168)(19 157 49 127)(20 176 50 146)(21 135 51 165)(22 154 52 124)(23 173 53 143)(24 132 54 162)(25 151 55 121)(26 170 56 140)(27 129 57 159)(28 148 58 178)(29 167 59 137)(30 126 60 156)(61 238 91 208)(62 197 92 227)(63 216 93 186)(64 235 94 205)(65 194 95 224)(66 213 96 183)(67 232 97 202)(68 191 98 221)(69 210 99 240)(70 229 100 199)(71 188 101 218)(72 207 102 237)(73 226 103 196)(74 185 104 215)(75 204 105 234)(76 223 106 193)(77 182 107 212)(78 201 108 231)(79 220 109 190)(80 239 110 209)(81 198 111 228)(82 217 112 187)(83 236 113 206)(84 195 114 225)(85 214 115 184)(86 233 116 203)(87 192 117 222)(88 211 118 181)(89 230 119 200)(90 189 120 219)
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), (1,160,236,98)(2,129,237,67)(3,158,238,96)(4,127,239,65)(5,156,240,94)(6,125,181,63)(7,154,182,92)(8,123,183,61)(9,152,184,90)(10,121,185,119)(11,150,186,88)(12,179,187,117)(13,148,188,86)(14,177,189,115)(15,146,190,84)(16,175,191,113)(17,144,192,82)(18,173,193,111)(19,142,194,80)(20,171,195,109)(21,140,196,78)(22,169,197,107)(23,138,198,76)(24,167,199,105)(25,136,200,74)(26,165,201,103)(27,134,202,72)(28,163,203,101)(29,132,204,70)(30,161,205,99)(31,130,206,68)(32,159,207,97)(33,128,208,66)(34,157,209,95)(35,126,210,64)(36,155,211,93)(37,124,212,62)(38,153,213,91)(39,122,214,120)(40,151,215,89)(41,180,216,118)(42,149,217,87)(43,178,218,116)(44,147,219,85)(45,176,220,114)(46,145,221,83)(47,174,222,112)(48,143,223,81)(49,172,224,110)(50,141,225,79)(51,170,226,108)(52,139,227,77)(53,168,228,106)(54,137,229,75)(55,166,230,104)(56,135,231,73)(57,164,232,102)(58,133,233,71)(59,162,234,100)(60,131,235,69), (1,175,31,145)(2,134,32,164)(3,153,33,123)(4,172,34,142)(5,131,35,161)(6,150,36,180)(7,169,37,139)(8,128,38,158)(9,147,39,177)(10,166,40,136)(11,125,41,155)(12,144,42,174)(13,163,43,133)(14,122,44,152)(15,141,45,171)(16,160,46,130)(17,179,47,149)(18,138,48,168)(19,157,49,127)(20,176,50,146)(21,135,51,165)(22,154,52,124)(23,173,53,143)(24,132,54,162)(25,151,55,121)(26,170,56,140)(27,129,57,159)(28,148,58,178)(29,167,59,137)(30,126,60,156)(61,238,91,208)(62,197,92,227)(63,216,93,186)(64,235,94,205)(65,194,95,224)(66,213,96,183)(67,232,97,202)(68,191,98,221)(69,210,99,240)(70,229,100,199)(71,188,101,218)(72,207,102,237)(73,226,103,196)(74,185,104,215)(75,204,105,234)(76,223,106,193)(77,182,107,212)(78,201,108,231)(79,220,109,190)(80,239,110,209)(81,198,111,228)(82,217,112,187)(83,236,113,206)(84,195,114,225)(85,214,115,184)(86,233,116,203)(87,192,117,222)(88,211,118,181)(89,230,119,200)(90,189,120,219)>;
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), (1,160,236,98)(2,129,237,67)(3,158,238,96)(4,127,239,65)(5,156,240,94)(6,125,181,63)(7,154,182,92)(8,123,183,61)(9,152,184,90)(10,121,185,119)(11,150,186,88)(12,179,187,117)(13,148,188,86)(14,177,189,115)(15,146,190,84)(16,175,191,113)(17,144,192,82)(18,173,193,111)(19,142,194,80)(20,171,195,109)(21,140,196,78)(22,169,197,107)(23,138,198,76)(24,167,199,105)(25,136,200,74)(26,165,201,103)(27,134,202,72)(28,163,203,101)(29,132,204,70)(30,161,205,99)(31,130,206,68)(32,159,207,97)(33,128,208,66)(34,157,209,95)(35,126,210,64)(36,155,211,93)(37,124,212,62)(38,153,213,91)(39,122,214,120)(40,151,215,89)(41,180,216,118)(42,149,217,87)(43,178,218,116)(44,147,219,85)(45,176,220,114)(46,145,221,83)(47,174,222,112)(48,143,223,81)(49,172,224,110)(50,141,225,79)(51,170,226,108)(52,139,227,77)(53,168,228,106)(54,137,229,75)(55,166,230,104)(56,135,231,73)(57,164,232,102)(58,133,233,71)(59,162,234,100)(60,131,235,69), (1,175,31,145)(2,134,32,164)(3,153,33,123)(4,172,34,142)(5,131,35,161)(6,150,36,180)(7,169,37,139)(8,128,38,158)(9,147,39,177)(10,166,40,136)(11,125,41,155)(12,144,42,174)(13,163,43,133)(14,122,44,152)(15,141,45,171)(16,160,46,130)(17,179,47,149)(18,138,48,168)(19,157,49,127)(20,176,50,146)(21,135,51,165)(22,154,52,124)(23,173,53,143)(24,132,54,162)(25,151,55,121)(26,170,56,140)(27,129,57,159)(28,148,58,178)(29,167,59,137)(30,126,60,156)(61,238,91,208)(62,197,92,227)(63,216,93,186)(64,235,94,205)(65,194,95,224)(66,213,96,183)(67,232,97,202)(68,191,98,221)(69,210,99,240)(70,229,100,199)(71,188,101,218)(72,207,102,237)(73,226,103,196)(74,185,104,215)(75,204,105,234)(76,223,106,193)(77,182,107,212)(78,201,108,231)(79,220,109,190)(80,239,110,209)(81,198,111,228)(82,217,112,187)(83,236,113,206)(84,195,114,225)(85,214,115,184)(86,233,116,203)(87,192,117,222)(88,211,118,181)(89,230,119,200)(90,189,120,219) );
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)], [(1,160,236,98),(2,129,237,67),(3,158,238,96),(4,127,239,65),(5,156,240,94),(6,125,181,63),(7,154,182,92),(8,123,183,61),(9,152,184,90),(10,121,185,119),(11,150,186,88),(12,179,187,117),(13,148,188,86),(14,177,189,115),(15,146,190,84),(16,175,191,113),(17,144,192,82),(18,173,193,111),(19,142,194,80),(20,171,195,109),(21,140,196,78),(22,169,197,107),(23,138,198,76),(24,167,199,105),(25,136,200,74),(26,165,201,103),(27,134,202,72),(28,163,203,101),(29,132,204,70),(30,161,205,99),(31,130,206,68),(32,159,207,97),(33,128,208,66),(34,157,209,95),(35,126,210,64),(36,155,211,93),(37,124,212,62),(38,153,213,91),(39,122,214,120),(40,151,215,89),(41,180,216,118),(42,149,217,87),(43,178,218,116),(44,147,219,85),(45,176,220,114),(46,145,221,83),(47,174,222,112),(48,143,223,81),(49,172,224,110),(50,141,225,79),(51,170,226,108),(52,139,227,77),(53,168,228,106),(54,137,229,75),(55,166,230,104),(56,135,231,73),(57,164,232,102),(58,133,233,71),(59,162,234,100),(60,131,235,69)], [(1,175,31,145),(2,134,32,164),(3,153,33,123),(4,172,34,142),(5,131,35,161),(6,150,36,180),(7,169,37,139),(8,128,38,158),(9,147,39,177),(10,166,40,136),(11,125,41,155),(12,144,42,174),(13,163,43,133),(14,122,44,152),(15,141,45,171),(16,160,46,130),(17,179,47,149),(18,138,48,168),(19,157,49,127),(20,176,50,146),(21,135,51,165),(22,154,52,124),(23,173,53,143),(24,132,54,162),(25,151,55,121),(26,170,56,140),(27,129,57,159),(28,148,58,178),(29,167,59,137),(30,126,60,156),(61,238,91,208),(62,197,92,227),(63,216,93,186),(64,235,94,205),(65,194,95,224),(66,213,96,183),(67,232,97,202),(68,191,98,221),(69,210,99,240),(70,229,100,199),(71,188,101,218),(72,207,102,237),(73,226,103,196),(74,185,104,215),(75,204,105,234),(76,223,106,193),(77,182,107,212),(78,201,108,231),(79,220,109,190),(80,239,110,209),(81,198,111,228),(82,217,112,187),(83,236,113,206),(84,195,114,225),(85,214,115,184),(86,233,116,203),(87,192,117,222),(88,211,118,181),(89,230,119,200),(90,189,120,219)]])
60 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 | 20B | 20C | 20D | 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 | 12 | 12 | 2 | 2 | 2 | 20 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C5⋊D4 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C15⋊D4 | C2×S3×D5 | D12⋊D5 |
| kernel | C60.89D4 | D6⋊Dic5 | C4×Dic15 | C6×Dic10 | C10×D12 | C2×Dic10 | C60 | C2×D12 | C2×Dic5 | C2×C20 | C30 | C2×C12 | C22×S3 | C20 | C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C60.89D4 ►in GL6(𝔽61)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 26 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 8 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 34 |
| 0 | 0 | 0 | 0 | 18 | 50 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 1 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 43 | 11 |
G:=sub<GL(6,GF(61))| [3,26,0,0,0,0,0,41,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[11,0,0,0,0,0,2,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,18,0,0,0,0,34,50],[11,1,0,0,0,0,2,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,43,0,0,0,0,0,11] >;
C60.89D4 in GAP, Magma, Sage, TeX
C_{60}._{89}D_4 % in TeX
G:=Group("C60.89D4"); // GroupNames label
G:=SmallGroup(480,446);
// by ID
G=gap.SmallGroup(480,446);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^29,c*a*c^-1=a^19,c*b*c^-1=a^30*b^-1>;
// generators/relations