metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.23D4, (C6×Q8)⋊4D5, (Q8×C10)⋊8S3, (Q8×C30)⋊4C2, (C2×Q8)⋊6D15, (C2×C4).57D30, (C4×Dic15)⋊7C2, (C2×D60).10C2, C30.390(C2×D4), (C2×C20).156D6, D30⋊3C4⋊38C2, (C2×C12).252D10, C5⋊4(C12.23D4), C20.48(C3⋊D4), C3⋊4(C20.23D4), C15⋊25(C4.4D4), C4.11(C15⋊7D4), C12.50(C5⋊D4), C30.262(C4○D4), C2.9(Q8⋊3D15), (C2×C60).436C22, (C2×C30).315C23, C6.46(Q8⋊2D5), C10.46(Q8⋊3S3), C22.65(C22×D15), (C22×D15).14C22, (C2×Dic15).175C22, C6.117(C2×C5⋊D4), C2.22(C2×C15⋊7D4), C10.117(C2×C3⋊D4), (C2×C6).311(C22×D5), (C2×C10).310(C22×S3), SmallGroup(480,912)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.23D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a-1, cbc=a30b-1 >
Subgroups: 1012 in 152 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D15, C30, C30, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, Dic15, C60, C60, D30, C2×C30, C4×Dic5, D10⋊C4, C2×D20, Q8×C10, C12.23D4, D60, C2×Dic15, C2×C60, C2×C60, Q8×C15, C22×D15, C20.23D4, C4×Dic15, D30⋊3C4, C2×D60, Q8×C30, C60.23D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, D15, C4.4D4, C5⋊D4, C22×D5, Q8⋊3S3, C2×C3⋊D4, D30, Q8⋊2D5, C2×C5⋊D4, C12.23D4, C15⋊7D4, C22×D15, C20.23D4, Q8⋊3D15, C2×C15⋊7D4, C60.23D4
►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)
(1 194 123 89)(2 223 124 118)(3 192 125 87)(4 221 126 116)(5 190 127 85)(6 219 128 114)(7 188 129 83)(8 217 130 112)(9 186 131 81)(10 215 132 110)(11 184 133 79)(12 213 134 108)(13 182 135 77)(14 211 136 106)(15 240 137 75)(16 209 138 104)(17 238 139 73)(18 207 140 102)(19 236 141 71)(20 205 142 100)(21 234 143 69)(22 203 144 98)(23 232 145 67)(24 201 146 96)(25 230 147 65)(26 199 148 94)(27 228 149 63)(28 197 150 92)(29 226 151 61)(30 195 152 90)(31 224 153 119)(32 193 154 88)(33 222 155 117)(34 191 156 86)(35 220 157 115)(36 189 158 84)(37 218 159 113)(38 187 160 82)(39 216 161 111)(40 185 162 80)(41 214 163 109)(42 183 164 78)(43 212 165 107)(44 181 166 76)(45 210 167 105)(46 239 168 74)(47 208 169 103)(48 237 170 72)(49 206 171 101)(50 235 172 70)(51 204 173 99)(52 233 174 68)(53 202 175 97)(54 231 176 66)(55 200 177 95)(56 229 178 64)(57 198 179 93)(58 227 180 62)(59 196 121 91)(60 225 122 120)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 150)(20 149)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 180)(50 179)(51 178)(52 177)(53 176)(54 175)(55 174)(56 173)(57 172)(58 171)(59 170)(60 169)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(181 222)(182 221)(183 220)(184 219)(185 218)(186 217)(187 216)(188 215)(189 214)(190 213)(191 212)(192 211)(193 210)(194 209)(195 208)(196 207)(197 206)(198 205)(199 204)(200 203)(201 202)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)
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), (1,194,123,89)(2,223,124,118)(3,192,125,87)(4,221,126,116)(5,190,127,85)(6,219,128,114)(7,188,129,83)(8,217,130,112)(9,186,131,81)(10,215,132,110)(11,184,133,79)(12,213,134,108)(13,182,135,77)(14,211,136,106)(15,240,137,75)(16,209,138,104)(17,238,139,73)(18,207,140,102)(19,236,141,71)(20,205,142,100)(21,234,143,69)(22,203,144,98)(23,232,145,67)(24,201,146,96)(25,230,147,65)(26,199,148,94)(27,228,149,63)(28,197,150,92)(29,226,151,61)(30,195,152,90)(31,224,153,119)(32,193,154,88)(33,222,155,117)(34,191,156,86)(35,220,157,115)(36,189,158,84)(37,218,159,113)(38,187,160,82)(39,216,161,111)(40,185,162,80)(41,214,163,109)(42,183,164,78)(43,212,165,107)(44,181,166,76)(45,210,167,105)(46,239,168,74)(47,208,169,103)(48,237,170,72)(49,206,171,101)(50,235,172,70)(51,204,173,99)(52,233,174,68)(53,202,175,97)(54,231,176,66)(55,200,177,95)(56,229,178,64)(57,198,179,93)(58,227,180,62)(59,196,121,91)(60,225,122,120), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,180)(50,179)(51,178)(52,177)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,203)(201,202)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232)>;
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), (1,194,123,89)(2,223,124,118)(3,192,125,87)(4,221,126,116)(5,190,127,85)(6,219,128,114)(7,188,129,83)(8,217,130,112)(9,186,131,81)(10,215,132,110)(11,184,133,79)(12,213,134,108)(13,182,135,77)(14,211,136,106)(15,240,137,75)(16,209,138,104)(17,238,139,73)(18,207,140,102)(19,236,141,71)(20,205,142,100)(21,234,143,69)(22,203,144,98)(23,232,145,67)(24,201,146,96)(25,230,147,65)(26,199,148,94)(27,228,149,63)(28,197,150,92)(29,226,151,61)(30,195,152,90)(31,224,153,119)(32,193,154,88)(33,222,155,117)(34,191,156,86)(35,220,157,115)(36,189,158,84)(37,218,159,113)(38,187,160,82)(39,216,161,111)(40,185,162,80)(41,214,163,109)(42,183,164,78)(43,212,165,107)(44,181,166,76)(45,210,167,105)(46,239,168,74)(47,208,169,103)(48,237,170,72)(49,206,171,101)(50,235,172,70)(51,204,173,99)(52,233,174,68)(53,202,175,97)(54,231,176,66)(55,200,177,95)(56,229,178,64)(57,198,179,93)(58,227,180,62)(59,196,121,91)(60,225,122,120), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,180)(50,179)(51,178)(52,177)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,203)(201,202)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232) );
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)], [(1,194,123,89),(2,223,124,118),(3,192,125,87),(4,221,126,116),(5,190,127,85),(6,219,128,114),(7,188,129,83),(8,217,130,112),(9,186,131,81),(10,215,132,110),(11,184,133,79),(12,213,134,108),(13,182,135,77),(14,211,136,106),(15,240,137,75),(16,209,138,104),(17,238,139,73),(18,207,140,102),(19,236,141,71),(20,205,142,100),(21,234,143,69),(22,203,144,98),(23,232,145,67),(24,201,146,96),(25,230,147,65),(26,199,148,94),(27,228,149,63),(28,197,150,92),(29,226,151,61),(30,195,152,90),(31,224,153,119),(32,193,154,88),(33,222,155,117),(34,191,156,86),(35,220,157,115),(36,189,158,84),(37,218,159,113),(38,187,160,82),(39,216,161,111),(40,185,162,80),(41,214,163,109),(42,183,164,78),(43,212,165,107),(44,181,166,76),(45,210,167,105),(46,239,168,74),(47,208,169,103),(48,237,170,72),(49,206,171,101),(50,235,172,70),(51,204,173,99),(52,233,174,68),(53,202,175,97),(54,231,176,66),(55,200,177,95),(56,229,178,64),(57,198,179,93),(58,227,180,62),(59,196,121,91),(60,225,122,120)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,156),(14,155),(15,154),(16,153),(17,152),(18,151),(19,150),(20,149),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,180),(50,179),(51,178),(52,177),(53,176),(54,175),(55,174),(56,173),(57,172),(58,171),(59,170),(60,169),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(181,222),(182,221),(183,220),(184,219),(185,218),(186,217),(187,216),(188,215),(189,214),(190,213),(191,212),(192,211),(193,210),(194,209),(195,208),(196,207),(197,206),(198,205),(199,204),(200,203),(201,202),(223,240),(224,239),(225,238),(226,237),(227,236),(228,235),(229,234),(230,233),(231,232)]])
84 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 | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | C4○D4 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | Q8⋊3S3 | Q8⋊2D5 | Q8⋊3D15 |
| kernel | C60.23D4 | C4×Dic15 | D30⋊3C4 | C2×D60 | Q8×C30 | Q8×C10 | C60 | C6×Q8 | C2×C20 | C30 | C2×C12 | C20 | C2×Q8 | C12 | C2×C4 | C4 | C10 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 4 | 4 | 8 | 12 | 16 | 2 | 4 | 8 |
Matrix representation of C60.23D4 ►in GL6(𝔽61)
| 18 | 18 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 31 | 0 | 0 |
| 0 | 0 | 30 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 8 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 30 | 44 | 0 | 0 | 0 | 0 |
| 53 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 33 | 0 | 0 |
| 0 | 0 | 14 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 27 |
| 0 | 0 | 0 | 0 | 43 | 11 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 28 | 0 | 0 |
| 0 | 0 | 47 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 53 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,47,30,0,0,0,0,31,25,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[30,53,0,0,0,0,44,31,0,0,0,0,0,0,24,14,0,0,0,0,33,37,0,0,0,0,0,0,50,43,0,0,0,0,27,11],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,37,47,0,0,0,0,28,24,0,0,0,0,0,0,1,0,0,0,0,0,53,60] >;
C60.23D4 in GAP, Magma, Sage, TeX
C_{60}._{23}D_4 % in TeX
G:=Group("C60.23D4"); // GroupNames label
G:=SmallGroup(480,912);
// by ID
G=gap.SmallGroup(480,912);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^-1,c*b*c=a^30*b^-1>;
// generators/relations