metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60.6C4, C40.74D6, C8.18D30, C24.79D10, Dic30.6C4, C60.254C23, C120.92C22, (C2×C8)⋊7D15, (C2×C40)⋊10S3, (C8×D15)⋊9C2, (C2×C24)⋊12D5, C5⋊7(C8○D12), (C2×C120)⋊16C2, C15⋊21(C8○D4), C20.85(C4×S3), (C2×C4).79D30, C12.53(C4×D5), C4.10(C4×D15), C15⋊7D4.6C4, C40⋊S3⋊15C2, C60.190(C2×C4), D30.24(C2×C4), (C2×C20).399D6, C60.7C4⋊23C2, C22.2(C4×D15), (C2×C12).404D10, C3⋊4(D20.3C4), C4.36(C22×D15), C30.165(C22×C4), (C2×C60).485C22, C20.224(C22×S3), C15⋊3C8.37C22, Dic15.31(C2×C4), (C4×D15).50C22, C12.226(C22×D5), D60⋊11C2.12C2, C6.70(C2×C4×D5), C2.15(C2×C4×D15), C10.102(S3×C2×C4), (C2×C6).33(C4×D5), (C2×C10).58(C4×S3), (C2×C30).140(C2×C4), SmallGroup(480,866)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60.6C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, ac=ca, bc=cb >
Subgroups: 596 in 124 conjugacy classes, 55 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, 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, C4.Dic3, C2×C24, C4○D12, Dic15, C60, D30, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, C8○D12, C15⋊3C8, C120, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20.3C4, C8×D15, C40⋊S3, C60.7C4, C2×C120, D60⋊11C2, D60.6C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, D15, C8○D4, C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, C8○D12, C4×D15, C22×D15, D20.3C4, C2×C4×D15, D60.6C4
►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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(121 126)(122 125)(123 124)(127 180)(128 179)(129 178)(130 177)(131 176)(132 175)(133 174)(134 173)(135 172)(136 171)(137 170)(138 169)(139 168)(140 167)(141 166)(142 165)(143 164)(144 163)(145 162)(146 161)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(181 236)(182 235)(183 234)(184 233)(185 232)(186 231)(187 230)(188 229)(189 228)(190 227)(191 226)(192 225)(193 224)(194 223)(195 222)(196 221)(197 220)(198 219)(199 218)(200 217)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(237 240)(238 239)
(1 154 100 209 31 124 70 239)(2 155 101 210 32 125 71 240)(3 156 102 211 33 126 72 181)(4 157 103 212 34 127 73 182)(5 158 104 213 35 128 74 183)(6 159 105 214 36 129 75 184)(7 160 106 215 37 130 76 185)(8 161 107 216 38 131 77 186)(9 162 108 217 39 132 78 187)(10 163 109 218 40 133 79 188)(11 164 110 219 41 134 80 189)(12 165 111 220 42 135 81 190)(13 166 112 221 43 136 82 191)(14 167 113 222 44 137 83 192)(15 168 114 223 45 138 84 193)(16 169 115 224 46 139 85 194)(17 170 116 225 47 140 86 195)(18 171 117 226 48 141 87 196)(19 172 118 227 49 142 88 197)(20 173 119 228 50 143 89 198)(21 174 120 229 51 144 90 199)(22 175 61 230 52 145 91 200)(23 176 62 231 53 146 92 201)(24 177 63 232 54 147 93 202)(25 178 64 233 55 148 94 203)(26 179 65 234 56 149 95 204)(27 180 66 235 57 150 96 205)(28 121 67 236 58 151 97 206)(29 122 68 237 59 152 98 207)(30 123 69 238 60 153 99 208)
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(121,126)(122,125)(123,124)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(181,236)(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)(188,229)(189,228)(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)(196,221)(197,220)(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(237,240)(238,239), (1,154,100,209,31,124,70,239)(2,155,101,210,32,125,71,240)(3,156,102,211,33,126,72,181)(4,157,103,212,34,127,73,182)(5,158,104,213,35,128,74,183)(6,159,105,214,36,129,75,184)(7,160,106,215,37,130,76,185)(8,161,107,216,38,131,77,186)(9,162,108,217,39,132,78,187)(10,163,109,218,40,133,79,188)(11,164,110,219,41,134,80,189)(12,165,111,220,42,135,81,190)(13,166,112,221,43,136,82,191)(14,167,113,222,44,137,83,192)(15,168,114,223,45,138,84,193)(16,169,115,224,46,139,85,194)(17,170,116,225,47,140,86,195)(18,171,117,226,48,141,87,196)(19,172,118,227,49,142,88,197)(20,173,119,228,50,143,89,198)(21,174,120,229,51,144,90,199)(22,175,61,230,52,145,91,200)(23,176,62,231,53,146,92,201)(24,177,63,232,54,147,93,202)(25,178,64,233,55,148,94,203)(26,179,65,234,56,149,95,204)(27,180,66,235,57,150,96,205)(28,121,67,236,58,151,97,206)(29,122,68,237,59,152,98,207)(30,123,69,238,60,153,99,208)>;
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(121,126)(122,125)(123,124)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(181,236)(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)(188,229)(189,228)(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)(196,221)(197,220)(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(237,240)(238,239), (1,154,100,209,31,124,70,239)(2,155,101,210,32,125,71,240)(3,156,102,211,33,126,72,181)(4,157,103,212,34,127,73,182)(5,158,104,213,35,128,74,183)(6,159,105,214,36,129,75,184)(7,160,106,215,37,130,76,185)(8,161,107,216,38,131,77,186)(9,162,108,217,39,132,78,187)(10,163,109,218,40,133,79,188)(11,164,110,219,41,134,80,189)(12,165,111,220,42,135,81,190)(13,166,112,221,43,136,82,191)(14,167,113,222,44,137,83,192)(15,168,114,223,45,138,84,193)(16,169,115,224,46,139,85,194)(17,170,116,225,47,140,86,195)(18,171,117,226,48,141,87,196)(19,172,118,227,49,142,88,197)(20,173,119,228,50,143,89,198)(21,174,120,229,51,144,90,199)(22,175,61,230,52,145,91,200)(23,176,62,231,53,146,92,201)(24,177,63,232,54,147,93,202)(25,178,64,233,55,148,94,203)(26,179,65,234,56,149,95,204)(27,180,66,235,57,150,96,205)(28,121,67,236,58,151,97,206)(29,122,68,237,59,152,98,207)(30,123,69,238,60,153,99,208) );
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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100),(121,126),(122,125),(123,124),(127,180),(128,179),(129,178),(130,177),(131,176),(132,175),(133,174),(134,173),(135,172),(136,171),(137,170),(138,169),(139,168),(140,167),(141,166),(142,165),(143,164),(144,163),(145,162),(146,161),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154),(181,236),(182,235),(183,234),(184,233),(185,232),(186,231),(187,230),(188,229),(189,228),(190,227),(191,226),(192,225),(193,224),(194,223),(195,222),(196,221),(197,220),(198,219),(199,218),(200,217),(201,216),(202,215),(203,214),(204,213),(205,212),(206,211),(207,210),(208,209),(237,240),(238,239)], [(1,154,100,209,31,124,70,239),(2,155,101,210,32,125,71,240),(3,156,102,211,33,126,72,181),(4,157,103,212,34,127,73,182),(5,158,104,213,35,128,74,183),(6,159,105,214,36,129,75,184),(7,160,106,215,37,130,76,185),(8,161,107,216,38,131,77,186),(9,162,108,217,39,132,78,187),(10,163,109,218,40,133,79,188),(11,164,110,219,41,134,80,189),(12,165,111,220,42,135,81,190),(13,166,112,221,43,136,82,191),(14,167,113,222,44,137,83,192),(15,168,114,223,45,138,84,193),(16,169,115,224,46,139,85,194),(17,170,116,225,47,140,86,195),(18,171,117,226,48,141,87,196),(19,172,118,227,49,142,88,197),(20,173,119,228,50,143,89,198),(21,174,120,229,51,144,90,199),(22,175,61,230,52,145,91,200),(23,176,62,231,53,146,92,201),(24,177,63,232,54,147,93,202),(25,178,64,233,55,148,94,203),(26,179,65,234,56,149,95,204),(27,180,66,235,57,150,96,205),(28,121,67,236,58,151,97,206),(29,122,68,237,59,152,98,207),(30,123,69,238,60,153,99,208)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 30 | 30 | 2 | 1 | 1 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×S3 | D15 | C8○D4 | C4×D5 | C4×D5 | D30 | D30 | C8○D12 | C4×D15 | C4×D15 | D20.3C4 | D60.6C4 |
| kernel | D60.6C4 | C8×D15 | C40⋊S3 | C60.7C4 | C2×C120 | D60⋊11C2 | Dic30 | D60 | C15⋊7D4 | C2×C40 | C2×C24 | C40 | C2×C20 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of D60.6C4 ►in GL4(𝔽241) generated by
| 131 | 80 | 0 | 0 |
| 161 | 148 | 0 | 0 |
| 0 | 0 | 185 | 44 |
| 0 | 0 | 115 | 0 |
| 157 | 147 | 0 | 0 |
| 93 | 84 | 0 | 0 |
| 0 | 0 | 0 | 197 |
| 0 | 0 | 115 | 0 |
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(241))| [131,161,0,0,80,148,0,0,0,0,185,115,0,0,44,0],[157,93,0,0,147,84,0,0,0,0,0,115,0,0,197,0],[177,0,0,0,0,177,0,0,0,0,8,0,0,0,0,8] >;
D60.6C4 in GAP, Magma, Sage, TeX
D_{60}._6C_4 % in TeX
G:=Group("D60.6C4"); // GroupNames label
G:=SmallGroup(480,866);
// by ID
G=gap.SmallGroup(480,866);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations