metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6)⋊1D20, (C2×C30)⋊8D4, C3⋊4(C20⋊7D4), C6.70(C2×D20), C15⋊23(C4⋊D4), (C5×Dic3)⋊16D4, C30.249(C2×D4), C10.164(S3×D4), D30⋊4C4⋊35C2, C23.33(S3×D5), C6.87(C4○D20), Dic3⋊5(C5⋊D4), C5⋊4(C23.14D6), C22⋊2(C3⋊D20), C6.Dic10⋊38C2, (C2×Dic5).65D6, (C22×Dic3)⋊7D5, (C22×D5).32D6, (C22×C6).45D10, C30.158(C4○D4), D10⋊Dic3⋊36C2, (C2×C30).211C23, (C22×C10).111D6, C10.59(D4⋊2S3), (C2×Dic3).166D10, (C22×C30).73C22, C2.30(Dic5.D6), (C6×Dic5).122C22, (C22×D15).70C22, (C2×Dic15).144C22, (C10×Dic3).204C22, (C2×C5⋊D4)⋊7S3, (C6×C5⋊D4)⋊7C2, (Dic3×C2×C10)⋊7C2, C6.68(C2×C5⋊D4), C2.44(S3×C5⋊D4), (C2×C3⋊D20)⋊15C2, (C2×C15⋊7D4)⋊19C2, C10.23(C2×C3⋊D4), C2.25(C2×C3⋊D20), (D5×C2×C6).55C22, (C2×C10)⋊12(C3⋊D4), C22.240(C2×S3×D5), (C2×C6).223(C22×D5), (C2×C10).223(C22×S3), SmallGroup(480,645)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6)⋊D20
G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, ac=ca, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1052 in 188 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C4⋊D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C6×D5, D30, C2×C30, C2×C30, C2×C30, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23.14D6, C3⋊D20, C6×Dic5, C3×C5⋊D4, C10×Dic3, C10×Dic3, C2×Dic15, C15⋊7D4, D5×C2×C6, C22×D15, C22×C30, C20⋊7D4, D10⋊Dic3, D30⋊4C4, C6.Dic10, C2×C3⋊D20, C6×C5⋊D4, Dic3×C2×C10, C2×C15⋊7D4, (C2×C6)⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, D20, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, C23.14D6, C3⋊D20, C2×S3×D5, C20⋊7D4, Dic5.D6, C2×C3⋊D20, S3×C5⋊D4, (C2×C6)⋊D20
►On 240 points
Generators in S240
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 174)(22 175)(23 176)(24 177)(25 178)(26 179)(27 180)(28 161)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 169)(37 170)(38 171)(39 172)(40 173)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 230)(202 231)(203 232)(204 233)(205 234)(206 235)(207 236)(208 237)(209 238)(210 239)(211 240)(212 221)(213 222)(214 223)(215 224)(216 225)(217 226)(218 227)(219 228)(220 229)
(1 82 140 198 107 152)(2 153 108 199 121 83)(3 84 122 200 109 154)(4 155 110 181 123 85)(5 86 124 182 111 156)(6 157 112 183 125 87)(7 88 126 184 113 158)(8 159 114 185 127 89)(9 90 128 186 115 160)(10 141 116 187 129 91)(11 92 130 188 117 142)(12 143 118 189 131 93)(13 94 132 190 119 144)(14 145 120 191 133 95)(15 96 134 192 101 146)(16 147 102 193 135 97)(17 98 136 194 103 148)(18 149 104 195 137 99)(19 100 138 196 105 150)(20 151 106 197 139 81)(21 79 220 164 60 239)(22 240 41 165 201 80)(23 61 202 166 42 221)(24 222 43 167 203 62)(25 63 204 168 44 223)(26 224 45 169 205 64)(27 65 206 170 46 225)(28 226 47 171 207 66)(29 67 208 172 48 227)(30 228 49 173 209 68)(31 69 210 174 50 229)(32 230 51 175 211 70)(33 71 212 176 52 231)(34 232 53 177 213 72)(35 73 214 178 54 233)(36 234 55 179 215 74)(37 75 216 180 56 235)(38 236 57 161 217 76)(39 77 218 162 58 237)(40 238 59 163 219 78)
(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 25)(2 24)(3 23)(4 22)(5 21)(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)(41 123)(42 122)(43 121)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 133)(52 132)(53 131)(54 130)(55 129)(56 128)(57 127)(58 126)(59 125)(60 124)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 160)(76 159)(77 158)(78 157)(79 156)(80 155)(81 224)(82 223)(83 222)(84 221)(85 240)(86 239)(87 238)(88 237)(89 236)(90 235)(91 234)(92 233)(93 232)(94 231)(95 230)(96 229)(97 228)(98 227)(99 226)(100 225)(101 210)(102 209)(103 208)(104 207)(105 206)(106 205)(107 204)(108 203)(109 202)(110 201)(111 220)(112 219)(113 218)(114 217)(115 216)(116 215)(117 214)(118 213)(119 212)(120 211)(161 185)(162 184)(163 183)(164 182)(165 181)(166 200)(167 199)(168 198)(169 197)(170 196)(171 195)(172 194)(173 193)(174 192)(175 191)(176 190)(177 189)(178 188)(179 187)(180 186)
G:=sub<Sym(240)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,230)(202,231)(203,232)(204,233)(205,234)(206,235)(207,236)(208,237)(209,238)(210,239)(211,240)(212,221)(213,222)(214,223)(215,224)(216,225)(217,226)(218,227)(219,228)(220,229), (1,82,140,198,107,152)(2,153,108,199,121,83)(3,84,122,200,109,154)(4,155,110,181,123,85)(5,86,124,182,111,156)(6,157,112,183,125,87)(7,88,126,184,113,158)(8,159,114,185,127,89)(9,90,128,186,115,160)(10,141,116,187,129,91)(11,92,130,188,117,142)(12,143,118,189,131,93)(13,94,132,190,119,144)(14,145,120,191,133,95)(15,96,134,192,101,146)(16,147,102,193,135,97)(17,98,136,194,103,148)(18,149,104,195,137,99)(19,100,138,196,105,150)(20,151,106,197,139,81)(21,79,220,164,60,239)(22,240,41,165,201,80)(23,61,202,166,42,221)(24,222,43,167,203,62)(25,63,204,168,44,223)(26,224,45,169,205,64)(27,65,206,170,46,225)(28,226,47,171,207,66)(29,67,208,172,48,227)(30,228,49,173,209,68)(31,69,210,174,50,229)(32,230,51,175,211,70)(33,71,212,176,52,231)(34,232,53,177,213,72)(35,73,214,178,54,233)(36,234,55,179,215,74)(37,75,216,180,56,235)(38,236,57,161,217,76)(39,77,218,162,58,237)(40,238,59,163,219,78), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(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)(41,123)(42,122)(43,121)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,132)(53,131)(54,130)(55,129)(56,128)(57,127)(58,126)(59,125)(60,124)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,224)(82,223)(83,222)(84,221)(85,240)(86,239)(87,238)(88,237)(89,236)(90,235)(91,234)(92,233)(93,232)(94,231)(95,230)(96,229)(97,228)(98,227)(99,226)(100,225)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,220)(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)(119,212)(120,211)(161,185)(162,184)(163,183)(164,182)(165,181)(166,200)(167,199)(168,198)(169,197)(170,196)(171,195)(172,194)(173,193)(174,192)(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,230)(202,231)(203,232)(204,233)(205,234)(206,235)(207,236)(208,237)(209,238)(210,239)(211,240)(212,221)(213,222)(214,223)(215,224)(216,225)(217,226)(218,227)(219,228)(220,229), (1,82,140,198,107,152)(2,153,108,199,121,83)(3,84,122,200,109,154)(4,155,110,181,123,85)(5,86,124,182,111,156)(6,157,112,183,125,87)(7,88,126,184,113,158)(8,159,114,185,127,89)(9,90,128,186,115,160)(10,141,116,187,129,91)(11,92,130,188,117,142)(12,143,118,189,131,93)(13,94,132,190,119,144)(14,145,120,191,133,95)(15,96,134,192,101,146)(16,147,102,193,135,97)(17,98,136,194,103,148)(18,149,104,195,137,99)(19,100,138,196,105,150)(20,151,106,197,139,81)(21,79,220,164,60,239)(22,240,41,165,201,80)(23,61,202,166,42,221)(24,222,43,167,203,62)(25,63,204,168,44,223)(26,224,45,169,205,64)(27,65,206,170,46,225)(28,226,47,171,207,66)(29,67,208,172,48,227)(30,228,49,173,209,68)(31,69,210,174,50,229)(32,230,51,175,211,70)(33,71,212,176,52,231)(34,232,53,177,213,72)(35,73,214,178,54,233)(36,234,55,179,215,74)(37,75,216,180,56,235)(38,236,57,161,217,76)(39,77,218,162,58,237)(40,238,59,163,219,78), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(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)(41,123)(42,122)(43,121)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,132)(53,131)(54,130)(55,129)(56,128)(57,127)(58,126)(59,125)(60,124)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,224)(82,223)(83,222)(84,221)(85,240)(86,239)(87,238)(88,237)(89,236)(90,235)(91,234)(92,233)(93,232)(94,231)(95,230)(96,229)(97,228)(98,227)(99,226)(100,225)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,220)(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)(119,212)(120,211)(161,185)(162,184)(163,183)(164,182)(165,181)(166,200)(167,199)(168,198)(169,197)(170,196)(171,195)(172,194)(173,193)(174,192)(175,191)(176,190)(177,189)(178,188)(179,187)(180,186) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,174),(22,175),(23,176),(24,177),(25,178),(26,179),(27,180),(28,161),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,168),(36,169),(37,170),(38,171),(39,172),(40,173),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199),(190,200),(201,230),(202,231),(203,232),(204,233),(205,234),(206,235),(207,236),(208,237),(209,238),(210,239),(211,240),(212,221),(213,222),(214,223),(215,224),(216,225),(217,226),(218,227),(219,228),(220,229)], [(1,82,140,198,107,152),(2,153,108,199,121,83),(3,84,122,200,109,154),(4,155,110,181,123,85),(5,86,124,182,111,156),(6,157,112,183,125,87),(7,88,126,184,113,158),(8,159,114,185,127,89),(9,90,128,186,115,160),(10,141,116,187,129,91),(11,92,130,188,117,142),(12,143,118,189,131,93),(13,94,132,190,119,144),(14,145,120,191,133,95),(15,96,134,192,101,146),(16,147,102,193,135,97),(17,98,136,194,103,148),(18,149,104,195,137,99),(19,100,138,196,105,150),(20,151,106,197,139,81),(21,79,220,164,60,239),(22,240,41,165,201,80),(23,61,202,166,42,221),(24,222,43,167,203,62),(25,63,204,168,44,223),(26,224,45,169,205,64),(27,65,206,170,46,225),(28,226,47,171,207,66),(29,67,208,172,48,227),(30,228,49,173,209,68),(31,69,210,174,50,229),(32,230,51,175,211,70),(33,71,212,176,52,231),(34,232,53,177,213,72),(35,73,214,178,54,233),(36,234,55,179,215,74),(37,75,216,180,56,235),(38,236,57,161,217,76),(39,77,218,162,58,237),(40,238,59,163,219,78)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(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),(41,123),(42,122),(43,121),(44,140),(45,139),(46,138),(47,137),(48,136),(49,135),(50,134),(51,133),(52,132),(53,131),(54,130),(55,129),(56,128),(57,127),(58,126),(59,125),(60,124),(61,154),(62,153),(63,152),(64,151),(65,150),(66,149),(67,148),(68,147),(69,146),(70,145),(71,144),(72,143),(73,142),(74,141),(75,160),(76,159),(77,158),(78,157),(79,156),(80,155),(81,224),(82,223),(83,222),(84,221),(85,240),(86,239),(87,238),(88,237),(89,236),(90,235),(91,234),(92,233),(93,232),(94,231),(95,230),(96,229),(97,228),(98,227),(99,226),(100,225),(101,210),(102,209),(103,208),(104,207),(105,206),(106,205),(107,204),(108,203),(109,202),(110,201),(111,220),(112,219),(113,218),(114,217),(115,216),(116,215),(117,214),(118,213),(119,212),(120,211),(161,185),(162,184),(163,183),(164,182),(165,181),(166,200),(167,199),(168,198),(169,197),(170,196),(171,195),(172,194),(173,193),(174,192),(175,191),(176,190),(177,189),(178,188),(179,187),(180,186)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 60 | 2 | 6 | 6 | 6 | 6 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C5⋊D4 | D20 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | C3⋊D20 | C2×S3×D5 | Dic5.D6 | S3×C5⋊D4 |
| kernel | (C2×C6)⋊D20 | D10⋊Dic3 | D30⋊4C4 | C6.Dic10 | C2×C3⋊D20 | C6×C5⋊D4 | Dic3×C2×C10 | C2×C15⋊7D4 | C2×C5⋊D4 | C5×Dic3 | C2×C30 | C22×Dic3 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | Dic3 | C2×C6 | C6 | C10 | C10 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of (C2×C6)⋊D20 ►in GL4(𝔽61) generated by
| 60 | 36 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 19 |
| 0 | 0 | 48 | 59 |
| 38 | 26 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 28 | 30 |
| 0 | 0 | 45 | 33 |
| 33 | 11 | 0 | 0 |
| 12 | 28 | 0 | 0 |
| 0 | 0 | 19 | 43 |
| 0 | 0 | 20 | 42 |
G:=sub<GL(4,GF(61))| [60,0,0,0,36,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,48,0,0,19,59],[38,0,0,0,26,53,0,0,0,0,28,45,0,0,30,33],[33,12,0,0,11,28,0,0,0,0,19,20,0,0,43,42] >;
(C2×C6)⋊D20 in GAP, Magma, Sage, TeX
(C_2\times C_6)\rtimes D_{20} % in TeX
G:=Group("(C2xC6):D20"); // GroupNames label
G:=SmallGroup(480,645);
// by ID
G=gap.SmallGroup(480,645);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations