metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60.4C4, Dic30.4C4, C60.184C23, C3⋊C8.35D10, C15⋊16(C8○D4), C20.66(C4×S3), C5⋊6(D12.C4), C12.21(C4×D5), C5⋊2C8.26D6, C15⋊7D4.5C4, C60.101(C2×C4), D30.21(C2×C4), (C2×C20).313D6, (C2×C12).82D10, C4.Dic5⋊11S3, D15⋊2C8⋊12C2, C3⋊3(D20.3C4), C4.4(D30.C2), (C2×C60).41C22, D30.5C4⋊13C2, D60⋊11C2.5C2, C20.181(C22×S3), C30.106(C22×C4), Dic15.22(C2×C4), (C4×D15).41C22, C12.181(C22×D5), C22.1(D30.C2), (C2×C3⋊C8)⋊4D5, (C10×C3⋊C8)⋊2C2, C6.40(C2×C4×D5), C10.73(S3×C2×C4), C4.154(C2×S3×D5), (C2×C6).9(C4×D5), (C2×C4).93(S3×D5), (C2×C10).45(C4×S3), (C5×C3⋊C8).40C22, (C3×C4.Dic5)⋊6C2, C2.5(C2×D30.C2), (C2×C30).103(C2×C4), (C3×C5⋊2C8).26C22, SmallGroup(480,367)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60.4C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a19, cbc-1=a18b >
Subgroups: 572 in 124 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, D15, C30, C30, C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, Dic15, C60, D30, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D12.C4, C5×C3⋊C8, C3×C5⋊2C8, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20.3C4, D15⋊2C8, D30.5C4, C3×C4.Dic5, C10×C3⋊C8, D60⋊11C2, D60.4C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C8○D4, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D12.C4, D30.C2, C2×S3×D5, D20.3C4, C2×D30.C2, D60.4C4
►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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(121 165)(122 164)(123 163)(124 162)(125 161)(126 160)(127 159)(128 158)(129 157)(130 156)(131 155)(132 154)(133 153)(134 152)(135 151)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(182 240)(183 239)(184 238)(185 237)(186 236)(187 235)(188 234)(189 233)(190 232)(191 231)(192 230)(193 229)(194 228)(195 227)(196 226)(197 225)(198 224)(199 223)(200 222)(201 221)(202 220)(203 219)(204 218)(205 217)(206 216)(207 215)(208 214)(209 213)(210 212)
(1 234 86 136 31 204 116 166)(2 193 87 155 32 223 117 125)(3 212 88 174 33 182 118 144)(4 231 89 133 34 201 119 163)(5 190 90 152 35 220 120 122)(6 209 91 171 36 239 61 141)(7 228 92 130 37 198 62 160)(8 187 93 149 38 217 63 179)(9 206 94 168 39 236 64 138)(10 225 95 127 40 195 65 157)(11 184 96 146 41 214 66 176)(12 203 97 165 42 233 67 135)(13 222 98 124 43 192 68 154)(14 181 99 143 44 211 69 173)(15 200 100 162 45 230 70 132)(16 219 101 121 46 189 71 151)(17 238 102 140 47 208 72 170)(18 197 103 159 48 227 73 129)(19 216 104 178 49 186 74 148)(20 235 105 137 50 205 75 167)(21 194 106 156 51 224 76 126)(22 213 107 175 52 183 77 145)(23 232 108 134 53 202 78 164)(24 191 109 153 54 221 79 123)(25 210 110 172 55 240 80 142)(26 229 111 131 56 199 81 161)(27 188 112 150 57 218 82 180)(28 207 113 169 58 237 83 139)(29 226 114 128 59 196 84 158)(30 185 115 147 60 215 85 177)
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,165)(122,164)(123,163)(124,162)(125,161)(126,160)(127,159)(128,158)(129,157)(130,156)(131,155)(132,154)(133,153)(134,152)(135,151)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(182,240)(183,239)(184,238)(185,237)(186,236)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,226)(197,225)(198,224)(199,223)(200,222)(201,221)(202,220)(203,219)(204,218)(205,217)(206,216)(207,215)(208,214)(209,213)(210,212), (1,234,86,136,31,204,116,166)(2,193,87,155,32,223,117,125)(3,212,88,174,33,182,118,144)(4,231,89,133,34,201,119,163)(5,190,90,152,35,220,120,122)(6,209,91,171,36,239,61,141)(7,228,92,130,37,198,62,160)(8,187,93,149,38,217,63,179)(9,206,94,168,39,236,64,138)(10,225,95,127,40,195,65,157)(11,184,96,146,41,214,66,176)(12,203,97,165,42,233,67,135)(13,222,98,124,43,192,68,154)(14,181,99,143,44,211,69,173)(15,200,100,162,45,230,70,132)(16,219,101,121,46,189,71,151)(17,238,102,140,47,208,72,170)(18,197,103,159,48,227,73,129)(19,216,104,178,49,186,74,148)(20,235,105,137,50,205,75,167)(21,194,106,156,51,224,76,126)(22,213,107,175,52,183,77,145)(23,232,108,134,53,202,78,164)(24,191,109,153,54,221,79,123)(25,210,110,172,55,240,80,142)(26,229,111,131,56,199,81,161)(27,188,112,150,57,218,82,180)(28,207,113,169,58,237,83,139)(29,226,114,128,59,196,84,158)(30,185,115,147,60,215,85,177)>;
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,165)(122,164)(123,163)(124,162)(125,161)(126,160)(127,159)(128,158)(129,157)(130,156)(131,155)(132,154)(133,153)(134,152)(135,151)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(182,240)(183,239)(184,238)(185,237)(186,236)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,226)(197,225)(198,224)(199,223)(200,222)(201,221)(202,220)(203,219)(204,218)(205,217)(206,216)(207,215)(208,214)(209,213)(210,212), (1,234,86,136,31,204,116,166)(2,193,87,155,32,223,117,125)(3,212,88,174,33,182,118,144)(4,231,89,133,34,201,119,163)(5,190,90,152,35,220,120,122)(6,209,91,171,36,239,61,141)(7,228,92,130,37,198,62,160)(8,187,93,149,38,217,63,179)(9,206,94,168,39,236,64,138)(10,225,95,127,40,195,65,157)(11,184,96,146,41,214,66,176)(12,203,97,165,42,233,67,135)(13,222,98,124,43,192,68,154)(14,181,99,143,44,211,69,173)(15,200,100,162,45,230,70,132)(16,219,101,121,46,189,71,151)(17,238,102,140,47,208,72,170)(18,197,103,159,48,227,73,129)(19,216,104,178,49,186,74,148)(20,235,105,137,50,205,75,167)(21,194,106,156,51,224,76,126)(22,213,107,175,52,183,77,145)(23,232,108,134,53,202,78,164)(24,191,109,153,54,221,79,123)(25,210,110,172,55,240,80,142)(26,229,111,131,56,199,81,161)(27,188,112,150,57,218,82,180)(28,207,113,169,58,237,83,139)(29,226,114,128,59,196,84,158)(30,185,115,147,60,215,85,177) );
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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(121,165),(122,164),(123,163),(124,162),(125,161),(126,160),(127,159),(128,158),(129,157),(130,156),(131,155),(132,154),(133,153),(134,152),(135,151),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(182,240),(183,239),(184,238),(185,237),(186,236),(187,235),(188,234),(189,233),(190,232),(191,231),(192,230),(193,229),(194,228),(195,227),(196,226),(197,225),(198,224),(199,223),(200,222),(201,221),(202,220),(203,219),(204,218),(205,217),(206,216),(207,215),(208,214),(209,213),(210,212)], [(1,234,86,136,31,204,116,166),(2,193,87,155,32,223,117,125),(3,212,88,174,33,182,118,144),(4,231,89,133,34,201,119,163),(5,190,90,152,35,220,120,122),(6,209,91,171,36,239,61,141),(7,228,92,130,37,198,62,160),(8,187,93,149,38,217,63,179),(9,206,94,168,39,236,64,138),(10,225,95,127,40,195,65,157),(11,184,96,146,41,214,66,176),(12,203,97,165,42,233,67,135),(13,222,98,124,43,192,68,154),(14,181,99,143,44,211,69,173),(15,200,100,162,45,230,70,132),(16,219,101,121,46,189,71,151),(17,238,102,140,47,208,72,170),(18,197,103,159,48,227,73,129),(19,216,104,178,49,186,74,148),(20,235,105,137,50,205,75,167),(21,194,106,156,51,224,76,126),(22,213,107,175,52,183,77,145),(23,232,108,134,53,202,78,164),(24,191,109,153,54,221,79,123),(25,210,110,172,55,240,80,142),(26,229,111,131,56,199,81,161),(27,188,112,150,57,218,82,180),(28,207,113,169,58,237,83,139),(29,226,114,128,59,196,84,158),(30,185,115,147,60,215,85,177)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | ··· | 10F | 12A | 12B | 12C | 15A | 15B | 20A | ··· | 20H | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 30 | 30 | 2 | 1 | 1 | 2 | 30 | 30 | 2 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×S3 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 | S3×D5 | D12.C4 | D30.C2 | C2×S3×D5 | D30.C2 | D60.4C4 |
| kernel | D60.4C4 | D15⋊2C8 | D30.5C4 | C3×C4.Dic5 | C10×C3⋊C8 | D60⋊11C2 | Dic30 | D60 | C15⋊7D4 | C4.Dic5 | C2×C3⋊C8 | C5⋊2C8 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C15 | C12 | C2×C6 | C3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of D60.4C4 ►in GL4(𝔽241) generated by
| 197 | 78 | 0 | 0 |
| 163 | 78 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 1 |
| 240 | 51 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 240 |
| 115 | 90 | 0 | 0 |
| 171 | 126 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
G:=sub<GL(4,GF(241))| [197,163,0,0,78,78,0,0,0,0,0,240,0,0,1,1],[240,0,0,0,51,1,0,0,0,0,1,0,0,0,240,240],[115,171,0,0,90,126,0,0,0,0,64,0,0,0,0,64] >;
D60.4C4 in GAP, Magma, Sage, TeX
D_{60}._4C_4 % in TeX
G:=Group("D60.4C4"); // GroupNames label
G:=SmallGroup(480,367);
// by ID
G=gap.SmallGroup(480,367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations