metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.70D4, C20.20D12, (C4×Dic5)⋊4S3, (C2×Dic6)⋊3D5, D30⋊4C4⋊9C2, C5⋊3(C42⋊7S3), (C10×Dic6)⋊6C2, (C12×Dic5)⋊4C2, (C2×D60).18C2, C30.117(C2×D4), (C2×C20).116D6, C10.58(C2×D12), C15⋊9(C4.4D4), C10.7(C4○D12), C30.41(C4○D4), (C2×C12).299D10, C4.10(C5⋊D12), C12.60(C5⋊D4), C3⋊1(C20.23D4), (C2×C30).65C23, C6.9(Q8⋊2D5), (C2×C60).143C22, (C2×Dic3).18D10, (C2×Dic5).165D6, C2.12(C12.28D10), (C10×Dic3).37C22, (C6×Dic5).187C22, (C22×D15).24C22, C6.12(C2×C5⋊D4), (C2×C4).156(S3×D5), C2.16(C2×C5⋊D12), C22.151(C2×S3×D5), (C2×C6).77(C22×D5), (C2×C10).77(C22×S3), SmallGroup(480,451)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.70D4
G = < a,b,c | a60=b4=c2=1, bab-1=a49, cac=a-1, cbc=a30b-1 >
Subgroups: 972 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, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C30, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C5×Dic3, C3×Dic5, C60, D30, C2×C30, C4×Dic5, D10⋊C4, C2×D20, Q8×C10, C42⋊7S3, C6×Dic5, C5×Dic6, C10×Dic3, D60, C2×C60, C22×D15, C20.23D4, D30⋊4C4, C12×Dic5, C10×Dic6, C2×D60, C60.70D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4.4D4, C5⋊D4, C22×D5, C2×D12, C4○D12, S3×D5, Q8⋊2D5, C2×C5⋊D4, C42⋊7S3, C5⋊D12, C2×S3×D5, C20.23D4, C12.28D10, C2×C5⋊D12, C60.70D4
►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)
(1 239 101 124)(2 228 102 173)(3 217 103 162)(4 206 104 151)(5 195 105 140)(6 184 106 129)(7 233 107 178)(8 222 108 167)(9 211 109 156)(10 200 110 145)(11 189 111 134)(12 238 112 123)(13 227 113 172)(14 216 114 161)(15 205 115 150)(16 194 116 139)(17 183 117 128)(18 232 118 177)(19 221 119 166)(20 210 120 155)(21 199 61 144)(22 188 62 133)(23 237 63 122)(24 226 64 171)(25 215 65 160)(26 204 66 149)(27 193 67 138)(28 182 68 127)(29 231 69 176)(30 220 70 165)(31 209 71 154)(32 198 72 143)(33 187 73 132)(34 236 74 121)(35 225 75 170)(36 214 76 159)(37 203 77 148)(38 192 78 137)(39 181 79 126)(40 230 80 175)(41 219 81 164)(42 208 82 153)(43 197 83 142)(44 186 84 131)(45 235 85 180)(46 224 86 169)(47 213 87 158)(48 202 88 147)(49 191 89 136)(50 240 90 125)(51 229 91 174)(52 218 92 163)(53 207 93 152)(54 196 94 141)(55 185 95 130)(56 234 96 179)(57 223 97 168)(58 212 98 157)(59 201 99 146)(60 190 100 135)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 120)(58 119)(59 118)(60 117)(121 172)(122 171)(123 170)(124 169)(125 168)(126 167)(127 166)(128 165)(129 164)(130 163)(131 162)(132 161)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(173 180)(174 179)(175 178)(176 177)(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), (1,239,101,124)(2,228,102,173)(3,217,103,162)(4,206,104,151)(5,195,105,140)(6,184,106,129)(7,233,107,178)(8,222,108,167)(9,211,109,156)(10,200,110,145)(11,189,111,134)(12,238,112,123)(13,227,113,172)(14,216,114,161)(15,205,115,150)(16,194,116,139)(17,183,117,128)(18,232,118,177)(19,221,119,166)(20,210,120,155)(21,199,61,144)(22,188,62,133)(23,237,63,122)(24,226,64,171)(25,215,65,160)(26,204,66,149)(27,193,67,138)(28,182,68,127)(29,231,69,176)(30,220,70,165)(31,209,71,154)(32,198,72,143)(33,187,73,132)(34,236,74,121)(35,225,75,170)(36,214,76,159)(37,203,77,148)(38,192,78,137)(39,181,79,126)(40,230,80,175)(41,219,81,164)(42,208,82,153)(43,197,83,142)(44,186,84,131)(45,235,85,180)(46,224,86,169)(47,213,87,158)(48,202,88,147)(49,191,89,136)(50,240,90,125)(51,229,91,174)(52,218,92,163)(53,207,93,152)(54,196,94,141)(55,185,95,130)(56,234,96,179)(57,223,97,168)(58,212,98,157)(59,201,99,146)(60,190,100,135), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,120)(58,119)(59,118)(60,117)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(173,180)(174,179)(175,178)(176,177)(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), (1,239,101,124)(2,228,102,173)(3,217,103,162)(4,206,104,151)(5,195,105,140)(6,184,106,129)(7,233,107,178)(8,222,108,167)(9,211,109,156)(10,200,110,145)(11,189,111,134)(12,238,112,123)(13,227,113,172)(14,216,114,161)(15,205,115,150)(16,194,116,139)(17,183,117,128)(18,232,118,177)(19,221,119,166)(20,210,120,155)(21,199,61,144)(22,188,62,133)(23,237,63,122)(24,226,64,171)(25,215,65,160)(26,204,66,149)(27,193,67,138)(28,182,68,127)(29,231,69,176)(30,220,70,165)(31,209,71,154)(32,198,72,143)(33,187,73,132)(34,236,74,121)(35,225,75,170)(36,214,76,159)(37,203,77,148)(38,192,78,137)(39,181,79,126)(40,230,80,175)(41,219,81,164)(42,208,82,153)(43,197,83,142)(44,186,84,131)(45,235,85,180)(46,224,86,169)(47,213,87,158)(48,202,88,147)(49,191,89,136)(50,240,90,125)(51,229,91,174)(52,218,92,163)(53,207,93,152)(54,196,94,141)(55,185,95,130)(56,234,96,179)(57,223,97,168)(58,212,98,157)(59,201,99,146)(60,190,100,135), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,120)(58,119)(59,118)(60,117)(121,172)(122,171)(123,170)(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(173,180)(174,179)(175,178)(176,177)(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)], [(1,239,101,124),(2,228,102,173),(3,217,103,162),(4,206,104,151),(5,195,105,140),(6,184,106,129),(7,233,107,178),(8,222,108,167),(9,211,109,156),(10,200,110,145),(11,189,111,134),(12,238,112,123),(13,227,113,172),(14,216,114,161),(15,205,115,150),(16,194,116,139),(17,183,117,128),(18,232,118,177),(19,221,119,166),(20,210,120,155),(21,199,61,144),(22,188,62,133),(23,237,63,122),(24,226,64,171),(25,215,65,160),(26,204,66,149),(27,193,67,138),(28,182,68,127),(29,231,69,176),(30,220,70,165),(31,209,71,154),(32,198,72,143),(33,187,73,132),(34,236,74,121),(35,225,75,170),(36,214,76,159),(37,203,77,148),(38,192,78,137),(39,181,79,126),(40,230,80,175),(41,219,81,164),(42,208,82,153),(43,197,83,142),(44,186,84,131),(45,235,85,180),(46,224,86,169),(47,213,87,158),(48,202,88,147),(49,191,89,136),(50,240,90,125),(51,229,91,174),(52,218,92,163),(53,207,93,152),(54,196,94,141),(55,185,95,130),(56,234,96,179),(57,223,97,168),(58,212,98,157),(59,201,99,146),(60,190,100,135)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,84),(34,83),(35,82),(36,81),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,120),(58,119),(59,118),(60,117),(121,172),(122,171),(123,170),(124,169),(125,168),(126,167),(127,166),(128,165),(129,164),(130,163),(131,162),(132,161),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(173,180),(174,179),(175,178),(176,177),(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)]])
66 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 | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 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 | D12 | C5⋊D4 | C4○D12 | S3×D5 | Q8⋊2D5 | C5⋊D12 | C2×S3×D5 | C12.28D10 |
| kernel | C60.70D4 | D30⋊4C4 | C12×Dic5 | C10×Dic6 | C2×D60 | C4×Dic5 | C60 | C2×Dic6 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C60.70D4 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 38 |
| 0 | 0 | 0 | 0 | 23 | 38 |
| 23 | 46 | 0 | 0 | 0 | 0 |
| 15 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 44 | 0 | 0 |
| 0 | 0 | 60 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 43 | 52 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 46 |
| 0 | 0 | 0 | 0 | 23 | 38 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,44,0,0,0,0,17,17,0,0,0,0,0,0,15,23,0,0,0,0,38,38],[23,15,0,0,0,0,46,38,0,0,0,0,0,0,17,60,0,0,0,0,44,44,0,0,0,0,0,0,9,43,0,0,0,0,18,52],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,44,1,0,0,0,0,17,17,0,0,0,0,0,0,23,23,0,0,0,0,46,38] >;
C60.70D4 in GAP, Magma, Sage, TeX
C_{60}._{70}D_4 % in TeX
G:=Group("C60.70D4"); // GroupNames label
G:=SmallGroup(480,451);
// by ID
G=gap.SmallGroup(480,451);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,120,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^49,c*a*c=a^-1,c*b*c=a^30*b^-1>;
// generators/relations