metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4⋊15D5, (C2×C20).1D6, D6⋊Dic5⋊1C2, C6.123(D4×D5), C30.98(C2×D4), D30⋊4C4⋊1C2, (C2×Dic10)⋊2S3, C6.1(C4○D20), C30.1(C4○D4), C15⋊1(C4.4D4), (C3×Dic5).5D4, D30⋊3C4⋊13C2, (Dic3×Dic5)⋊1C2, (C6×Dic10)⋊14C2, C6.1(D4⋊2D5), (C2×C12).215D10, C5⋊2(C12.23D4), (C2×C30).16C23, (C2×Dic5).80D6, (C22×S3).1D10, C2.6(D60⋊C2), C2.6(D12⋊D5), (C2×C60).246C22, C10.1(Q8⋊3S3), (C2×Dic3).72D10, Dic5.9(C3⋊D4), C3⋊6(Dic5.5D4), (C6×Dic5).3C22, (C10×Dic3).3C22, (C2×Dic15).24C22, (C22×D15).15C22, C2.8(D5×C3⋊D4), (C5×D6⋊C4)⋊15C2, (C2×C4).19(S3×D5), (C2×C5⋊D12).1C2, C10.25(C2×C3⋊D4), (S3×C2×C10).1C22, C22.113(C2×S3×D5), (C2×C6).28(C22×D5), (C2×C10).28(C22×S3), SmallGroup(480,402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).D6
G = < a,b,c,d | a2=b20=1, c6=b10, d2=a, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=c5 >
Subgroups: 860 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C5×S3, D15, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4, C2×D12, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, D30, C2×C30, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C12.23D4, C5⋊D12, C3×Dic10, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic5.5D4, Dic3×Dic5, D6⋊Dic5, D30⋊4C4, C5×D6⋊C4, D30⋊3C4, C2×C5⋊D12, C6×Dic10, (C2×C20).D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4.4D4, C22×D5, Q8⋊3S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D4⋊2D5, C12.23D4, C2×S3×D5, Dic5.5D4, D12⋊D5, D60⋊C2, D5×C3⋊D4, (C2×C20).D6
►On 240 points
Generators in S240
(1 197)(2 198)(3 199)(4 200)(5 181)(6 182)(7 183)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 190)(15 191)(16 192)(17 193)(18 194)(19 195)(20 196)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 101)(38 102)(39 103)(40 104)(41 159)(42 160)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(121 223)(122 224)(123 225)(124 226)(125 227)(126 228)(127 229)(128 230)(129 231)(130 232)(131 233)(132 234)(133 235)(134 236)(135 237)(136 238)(137 239)(138 240)(139 221)(140 222)(161 202)(162 203)(163 204)(164 205)(165 206)(166 207)(167 208)(168 209)(169 210)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 217)(177 218)(178 219)(179 220)(180 201)
(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)
(1 101 212 68 59 135 11 111 202 78 49 125)(2 120 213 67 60 134 12 110 203 77 50 124)(3 119 214 66 41 133 13 109 204 76 51 123)(4 118 215 65 42 132 14 108 205 75 52 122)(5 117 216 64 43 131 15 107 206 74 53 121)(6 116 217 63 44 130 16 106 207 73 54 140)(7 115 218 62 45 129 17 105 208 72 55 139)(8 114 219 61 46 128 18 104 209 71 56 138)(9 113 220 80 47 127 19 103 210 70 57 137)(10 112 201 79 48 126 20 102 211 69 58 136)(21 167 82 153 221 183 31 177 92 143 231 193)(22 166 83 152 222 182 32 176 93 142 232 192)(23 165 84 151 223 181 33 175 94 141 233 191)(24 164 85 150 224 200 34 174 95 160 234 190)(25 163 86 149 225 199 35 173 96 159 235 189)(26 162 87 148 226 198 36 172 97 158 236 188)(27 161 88 147 227 197 37 171 98 157 237 187)(28 180 89 146 228 196 38 170 99 156 238 186)(29 179 90 145 229 195 39 169 100 155 239 185)(30 178 91 144 230 194 40 168 81 154 240 184)
(1 217 197 176)(2 175 198 216)(3 215 199 174)(4 173 200 214)(5 213 181 172)(6 171 182 212)(7 211 183 170)(8 169 184 210)(9 209 185 168)(10 167 186 208)(11 207 187 166)(12 165 188 206)(13 205 189 164)(14 163 190 204)(15 203 191 162)(16 161 192 202)(17 201 193 180)(18 179 194 220)(19 219 195 178)(20 177 196 218)(21 38 105 102)(22 101 106 37)(23 36 107 120)(24 119 108 35)(25 34 109 118)(26 117 110 33)(27 32 111 116)(28 115 112 31)(29 30 113 114)(39 40 103 104)(41 52 159 150)(42 149 160 51)(43 50 141 148)(44 147 142 49)(45 48 143 146)(46 145 144 47)(53 60 151 158)(54 157 152 59)(55 58 153 156)(56 155 154 57)(61 239 91 137)(62 136 92 238)(63 237 93 135)(64 134 94 236)(65 235 95 133)(66 132 96 234)(67 233 97 131)(68 130 98 232)(69 231 99 129)(70 128 100 230)(71 229 81 127)(72 126 82 228)(73 227 83 125)(74 124 84 226)(75 225 85 123)(76 122 86 224)(77 223 87 121)(78 140 88 222)(79 221 89 139)(80 138 90 240)
G:=sub<Sym(240)| (1,197)(2,198)(3,199)(4,200)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,223)(122,224)(123,225)(124,226)(125,227)(126,228)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,221)(140,222)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,201), (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), (1,101,212,68,59,135,11,111,202,78,49,125)(2,120,213,67,60,134,12,110,203,77,50,124)(3,119,214,66,41,133,13,109,204,76,51,123)(4,118,215,65,42,132,14,108,205,75,52,122)(5,117,216,64,43,131,15,107,206,74,53,121)(6,116,217,63,44,130,16,106,207,73,54,140)(7,115,218,62,45,129,17,105,208,72,55,139)(8,114,219,61,46,128,18,104,209,71,56,138)(9,113,220,80,47,127,19,103,210,70,57,137)(10,112,201,79,48,126,20,102,211,69,58,136)(21,167,82,153,221,183,31,177,92,143,231,193)(22,166,83,152,222,182,32,176,93,142,232,192)(23,165,84,151,223,181,33,175,94,141,233,191)(24,164,85,150,224,200,34,174,95,160,234,190)(25,163,86,149,225,199,35,173,96,159,235,189)(26,162,87,148,226,198,36,172,97,158,236,188)(27,161,88,147,227,197,37,171,98,157,237,187)(28,180,89,146,228,196,38,170,99,156,238,186)(29,179,90,145,229,195,39,169,100,155,239,185)(30,178,91,144,230,194,40,168,81,154,240,184), (1,217,197,176)(2,175,198,216)(3,215,199,174)(4,173,200,214)(5,213,181,172)(6,171,182,212)(7,211,183,170)(8,169,184,210)(9,209,185,168)(10,167,186,208)(11,207,187,166)(12,165,188,206)(13,205,189,164)(14,163,190,204)(15,203,191,162)(16,161,192,202)(17,201,193,180)(18,179,194,220)(19,219,195,178)(20,177,196,218)(21,38,105,102)(22,101,106,37)(23,36,107,120)(24,119,108,35)(25,34,109,118)(26,117,110,33)(27,32,111,116)(28,115,112,31)(29,30,113,114)(39,40,103,104)(41,52,159,150)(42,149,160,51)(43,50,141,148)(44,147,142,49)(45,48,143,146)(46,145,144,47)(53,60,151,158)(54,157,152,59)(55,58,153,156)(56,155,154,57)(61,239,91,137)(62,136,92,238)(63,237,93,135)(64,134,94,236)(65,235,95,133)(66,132,96,234)(67,233,97,131)(68,130,98,232)(69,231,99,129)(70,128,100,230)(71,229,81,127)(72,126,82,228)(73,227,83,125)(74,124,84,226)(75,225,85,123)(76,122,86,224)(77,223,87,121)(78,140,88,222)(79,221,89,139)(80,138,90,240)>;
G:=Group( (1,197)(2,198)(3,199)(4,200)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,223)(122,224)(123,225)(124,226)(125,227)(126,228)(127,229)(128,230)(129,231)(130,232)(131,233)(132,234)(133,235)(134,236)(135,237)(136,238)(137,239)(138,240)(139,221)(140,222)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,201), (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), (1,101,212,68,59,135,11,111,202,78,49,125)(2,120,213,67,60,134,12,110,203,77,50,124)(3,119,214,66,41,133,13,109,204,76,51,123)(4,118,215,65,42,132,14,108,205,75,52,122)(5,117,216,64,43,131,15,107,206,74,53,121)(6,116,217,63,44,130,16,106,207,73,54,140)(7,115,218,62,45,129,17,105,208,72,55,139)(8,114,219,61,46,128,18,104,209,71,56,138)(9,113,220,80,47,127,19,103,210,70,57,137)(10,112,201,79,48,126,20,102,211,69,58,136)(21,167,82,153,221,183,31,177,92,143,231,193)(22,166,83,152,222,182,32,176,93,142,232,192)(23,165,84,151,223,181,33,175,94,141,233,191)(24,164,85,150,224,200,34,174,95,160,234,190)(25,163,86,149,225,199,35,173,96,159,235,189)(26,162,87,148,226,198,36,172,97,158,236,188)(27,161,88,147,227,197,37,171,98,157,237,187)(28,180,89,146,228,196,38,170,99,156,238,186)(29,179,90,145,229,195,39,169,100,155,239,185)(30,178,91,144,230,194,40,168,81,154,240,184), (1,217,197,176)(2,175,198,216)(3,215,199,174)(4,173,200,214)(5,213,181,172)(6,171,182,212)(7,211,183,170)(8,169,184,210)(9,209,185,168)(10,167,186,208)(11,207,187,166)(12,165,188,206)(13,205,189,164)(14,163,190,204)(15,203,191,162)(16,161,192,202)(17,201,193,180)(18,179,194,220)(19,219,195,178)(20,177,196,218)(21,38,105,102)(22,101,106,37)(23,36,107,120)(24,119,108,35)(25,34,109,118)(26,117,110,33)(27,32,111,116)(28,115,112,31)(29,30,113,114)(39,40,103,104)(41,52,159,150)(42,149,160,51)(43,50,141,148)(44,147,142,49)(45,48,143,146)(46,145,144,47)(53,60,151,158)(54,157,152,59)(55,58,153,156)(56,155,154,57)(61,239,91,137)(62,136,92,238)(63,237,93,135)(64,134,94,236)(65,235,95,133)(66,132,96,234)(67,233,97,131)(68,130,98,232)(69,231,99,129)(70,128,100,230)(71,229,81,127)(72,126,82,228)(73,227,83,125)(74,124,84,226)(75,225,85,123)(76,122,86,224)(77,223,87,121)(78,140,88,222)(79,221,89,139)(80,138,90,240) );
G=PermutationGroup([[(1,197),(2,198),(3,199),(4,200),(5,181),(6,182),(7,183),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,190),(15,191),(16,192),(17,193),(18,194),(19,195),(20,196),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,101),(38,102),(39,103),(40,104),(41,159),(42,160),(43,141),(44,142),(45,143),(46,144),(47,145),(48,146),(49,147),(50,148),(51,149),(52,150),(53,151),(54,152),(55,153),(56,154),(57,155),(58,156),(59,157),(60,158),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(121,223),(122,224),(123,225),(124,226),(125,227),(126,228),(127,229),(128,230),(129,231),(130,232),(131,233),(132,234),(133,235),(134,236),(135,237),(136,238),(137,239),(138,240),(139,221),(140,222),(161,202),(162,203),(163,204),(164,205),(165,206),(166,207),(167,208),(168,209),(169,210),(170,211),(171,212),(172,213),(173,214),(174,215),(175,216),(176,217),(177,218),(178,219),(179,220),(180,201)], [(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)], [(1,101,212,68,59,135,11,111,202,78,49,125),(2,120,213,67,60,134,12,110,203,77,50,124),(3,119,214,66,41,133,13,109,204,76,51,123),(4,118,215,65,42,132,14,108,205,75,52,122),(5,117,216,64,43,131,15,107,206,74,53,121),(6,116,217,63,44,130,16,106,207,73,54,140),(7,115,218,62,45,129,17,105,208,72,55,139),(8,114,219,61,46,128,18,104,209,71,56,138),(9,113,220,80,47,127,19,103,210,70,57,137),(10,112,201,79,48,126,20,102,211,69,58,136),(21,167,82,153,221,183,31,177,92,143,231,193),(22,166,83,152,222,182,32,176,93,142,232,192),(23,165,84,151,223,181,33,175,94,141,233,191),(24,164,85,150,224,200,34,174,95,160,234,190),(25,163,86,149,225,199,35,173,96,159,235,189),(26,162,87,148,226,198,36,172,97,158,236,188),(27,161,88,147,227,197,37,171,98,157,237,187),(28,180,89,146,228,196,38,170,99,156,238,186),(29,179,90,145,229,195,39,169,100,155,239,185),(30,178,91,144,230,194,40,168,81,154,240,184)], [(1,217,197,176),(2,175,198,216),(3,215,199,174),(4,173,200,214),(5,213,181,172),(6,171,182,212),(7,211,183,170),(8,169,184,210),(9,209,185,168),(10,167,186,208),(11,207,187,166),(12,165,188,206),(13,205,189,164),(14,163,190,204),(15,203,191,162),(16,161,192,202),(17,201,193,180),(18,179,194,220),(19,219,195,178),(20,177,196,218),(21,38,105,102),(22,101,106,37),(23,36,107,120),(24,119,108,35),(25,34,109,118),(26,117,110,33),(27,32,111,116),(28,115,112,31),(29,30,113,114),(39,40,103,104),(41,52,159,150),(42,149,160,51),(43,50,141,148),(44,147,142,49),(45,48,143,146),(46,145,144,47),(53,60,151,158),(54,157,152,59),(55,58,153,156),(56,155,154,57),(61,239,91,137),(62,136,92,238),(63,237,93,135),(64,134,94,236),(65,235,95,133),(66,132,96,234),(67,233,97,131),(68,130,98,232),(69,231,99,129),(70,128,100,230),(71,229,81,127),(72,126,82,228),(73,227,83,125),(74,124,84,226),(75,225,85,123),(76,122,86,224),(77,223,87,121),(78,140,88,222),(79,221,89,139),(80,138,90,240)]])
60 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 | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 60 | 2 | 4 | 6 | 6 | 10 | 10 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C4○D20 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊D5 | D60⋊C2 | D5×C3⋊D4 |
| kernel | (C2×C20).D6 | Dic3×Dic5 | D6⋊Dic5 | D30⋊4C4 | C5×D6⋊C4 | D30⋊3C4 | C2×C5⋊D12 | C6×Dic10 | C2×Dic10 | C3×Dic5 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | C6 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C2×C20).D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 50 | 60 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 36 | 0 | 0 |
| 0 | 0 | 25 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 50 | 0 | 0 | 0 | 0 |
| 39 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 53 | 0 | 0 |
| 0 | 0 | 31 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 36 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,0,0,0,0,0,60,11,0,0,0,0,0,0,57,25,0,0,0,0,36,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,39,0,0,0,0,50,60,0,0,0,0,0,0,53,31,0,0,0,0,53,8,0,0,0,0,0,0,0,60,0,0,0,0,1,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,34,36,0,0,0,0,34,27,0,0,0,0,0,0,1,0,0,0,0,0,60,60] >;
(C2×C20).D6 in GAP, Magma, Sage, TeX
(C_2\times C_{20}).D_6 % in TeX
G:=Group("(C2xC20).D6"); // GroupNames label
G:=SmallGroup(480,402);
// by ID
G=gap.SmallGroup(480,402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=1,c^6=b^10,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=c^5>;
// generators/relations