G = C6.(C2×D20) order 480 = 25·3·5
20th non-split extension by C6 of C2×D20 acting via C2×D20/C22×D5=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6).8D20, C6.65(C2×D20), (C2×C30).68D4, C6.D4⋊6D5, C30.217(C2×D4), C23.49(S3×D5), C6.Dic10⋊30C2, (C2×Dic5).56D6, (C22×D5).29D6, (C22×C6).26D10, (C22×C10).40D6, C30.138(C4○D4), C6.79(D4⋊2D5), D10⋊Dic3⋊28C2, (C2×C30).179C23, (C2×Dic3).55D10, C3⋊5(C22.D20), C5⋊2(C23.23D6), C10.79(D4⋊2S3), (C22×Dic15)⋊11C2, C22.11(C3⋊D20), C15⋊17(C22.D4), (C22×C30).41C22, C2.24(C30.C23), (C6×Dic5).106C22, (C2×Dic15).222C22, (C10×Dic3).105C22, (C6×C5⋊D4).4C2, (C2×C5⋊D4).4S3, C10.19(C2×C3⋊D4), C2.22(C2×C3⋊D20), (D5×C2×C6).47C22, C22.222(C2×S3×D5), (C5×C6.D4)⋊6C2, (C2×C10).15(C3⋊D4), (C2×C6).191(C22×D5), (C2×C10).191(C22×S3), SmallGroup(480,613)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.(C2×D20)
G = < a,b,c,d | a6=b2=c20=1, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=a3b, bd=db, dcd-1=a3c-1 >
Subgroups: 732 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, 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, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C6.D4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C2×C30, C2×C30, C2×C30, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C23.23D6, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, C2×Dic15, D5×C2×C6, C22×C30, C22.D20, D10⋊Dic3, C6.Dic10, C5×C6.D4, C6×C5⋊D4, C22×Dic15, C6.(C2×D20)
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, D20, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C2×D20, D4⋊2D5, C23.23D6, C3⋊D20, C2×S3×D5, C22.D20, C30.C23, C2×C3⋊D20, C6.(C2×D20)
►On 240 points
Generators in S240
(1 70 161 85 210 102)(2 103 211 86 162 71)(3 72 163 87 212 104)(4 105 213 88 164 73)(5 74 165 89 214 106)(6 107 215 90 166 75)(7 76 167 91 216 108)(8 109 217 92 168 77)(9 78 169 93 218 110)(10 111 219 94 170 79)(11 80 171 95 220 112)(12 113 201 96 172 61)(13 62 173 97 202 114)(14 115 203 98 174 63)(15 64 175 99 204 116)(16 117 205 100 176 65)(17 66 177 81 206 118)(18 119 207 82 178 67)(19 68 179 83 208 120)(20 101 209 84 180 69)(21 182 227 123 56 160)(22 141 57 124 228 183)(23 184 229 125 58 142)(24 143 59 126 230 185)(25 186 231 127 60 144)(26 145 41 128 232 187)(27 188 233 129 42 146)(28 147 43 130 234 189)(29 190 235 131 44 148)(30 149 45 132 236 191)(31 192 237 133 46 150)(32 151 47 134 238 193)(33 194 239 135 48 152)(34 153 49 136 240 195)(35 196 221 137 50 154)(36 155 51 138 222 197)(37 198 223 139 52 156)(38 157 53 140 224 199)(39 200 225 121 54 158)(40 159 55 122 226 181)
(1 95)(2 12)(3 97)(4 14)(5 99)(6 16)(7 81)(8 18)(9 83)(10 20)(11 85)(13 87)(15 89)(17 91)(19 93)(21 31)(22 134)(23 33)(24 136)(25 35)(26 138)(27 37)(28 140)(29 39)(30 122)(32 124)(34 126)(36 128)(38 130)(40 132)(41 197)(42 52)(43 199)(44 54)(45 181)(46 56)(47 183)(48 58)(49 185)(50 60)(51 187)(53 189)(55 191)(57 193)(59 195)(61 71)(62 212)(63 73)(64 214)(65 75)(66 216)(67 77)(68 218)(69 79)(70 220)(72 202)(74 204)(76 206)(78 208)(80 210)(82 92)(84 94)(86 96)(88 98)(90 100)(101 111)(102 171)(103 113)(104 173)(105 115)(106 175)(107 117)(108 177)(109 119)(110 179)(112 161)(114 163)(116 165)(118 167)(120 169)(121 131)(123 133)(125 135)(127 137)(129 139)(141 238)(142 152)(143 240)(144 154)(145 222)(146 156)(147 224)(148 158)(149 226)(150 160)(151 228)(153 230)(155 232)(157 234)(159 236)(162 172)(164 174)(166 176)(168 178)(170 180)(182 192)(184 194)(186 196)(188 198)(190 200)(201 211)(203 213)(205 215)(207 217)(209 219)(221 231)(223 233)(225 235)(227 237)(229 239)
(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 47 85 193)(2 192 86 46)(3 45 87 191)(4 190 88 44)(5 43 89 189)(6 188 90 42)(7 41 91 187)(8 186 92 60)(9 59 93 185)(10 184 94 58)(11 57 95 183)(12 182 96 56)(13 55 97 181)(14 200 98 54)(15 53 99 199)(16 198 100 52)(17 51 81 197)(18 196 82 50)(19 49 83 195)(20 194 84 48)(21 172 123 113)(22 112 124 171)(23 170 125 111)(24 110 126 169)(25 168 127 109)(26 108 128 167)(27 166 129 107)(28 106 130 165)(29 164 131 105)(30 104 132 163)(31 162 133 103)(32 102 134 161)(33 180 135 101)(34 120 136 179)(35 178 137 119)(36 118 138 177)(37 176 139 117)(38 116 140 175)(39 174 121 115)(40 114 122 173)(61 227 201 160)(62 159 202 226)(63 225 203 158)(64 157 204 224)(65 223 205 156)(66 155 206 222)(67 221 207 154)(68 153 208 240)(69 239 209 152)(70 151 210 238)(71 237 211 150)(72 149 212 236)(73 235 213 148)(74 147 214 234)(75 233 215 146)(76 145 216 232)(77 231 217 144)(78 143 218 230)(79 229 219 142)(80 141 220 228)
G:=sub<Sym(240)| (1,70,161,85,210,102)(2,103,211,86,162,71)(3,72,163,87,212,104)(4,105,213,88,164,73)(5,74,165,89,214,106)(6,107,215,90,166,75)(7,76,167,91,216,108)(8,109,217,92,168,77)(9,78,169,93,218,110)(10,111,219,94,170,79)(11,80,171,95,220,112)(12,113,201,96,172,61)(13,62,173,97,202,114)(14,115,203,98,174,63)(15,64,175,99,204,116)(16,117,205,100,176,65)(17,66,177,81,206,118)(18,119,207,82,178,67)(19,68,179,83,208,120)(20,101,209,84,180,69)(21,182,227,123,56,160)(22,141,57,124,228,183)(23,184,229,125,58,142)(24,143,59,126,230,185)(25,186,231,127,60,144)(26,145,41,128,232,187)(27,188,233,129,42,146)(28,147,43,130,234,189)(29,190,235,131,44,148)(30,149,45,132,236,191)(31,192,237,133,46,150)(32,151,47,134,238,193)(33,194,239,135,48,152)(34,153,49,136,240,195)(35,196,221,137,50,154)(36,155,51,138,222,197)(37,198,223,139,52,156)(38,157,53,140,224,199)(39,200,225,121,54,158)(40,159,55,122,226,181), (1,95)(2,12)(3,97)(4,14)(5,99)(6,16)(7,81)(8,18)(9,83)(10,20)(11,85)(13,87)(15,89)(17,91)(19,93)(21,31)(22,134)(23,33)(24,136)(25,35)(26,138)(27,37)(28,140)(29,39)(30,122)(32,124)(34,126)(36,128)(38,130)(40,132)(41,197)(42,52)(43,199)(44,54)(45,181)(46,56)(47,183)(48,58)(49,185)(50,60)(51,187)(53,189)(55,191)(57,193)(59,195)(61,71)(62,212)(63,73)(64,214)(65,75)(66,216)(67,77)(68,218)(69,79)(70,220)(72,202)(74,204)(76,206)(78,208)(80,210)(82,92)(84,94)(86,96)(88,98)(90,100)(101,111)(102,171)(103,113)(104,173)(105,115)(106,175)(107,117)(108,177)(109,119)(110,179)(112,161)(114,163)(116,165)(118,167)(120,169)(121,131)(123,133)(125,135)(127,137)(129,139)(141,238)(142,152)(143,240)(144,154)(145,222)(146,156)(147,224)(148,158)(149,226)(150,160)(151,228)(153,230)(155,232)(157,234)(159,236)(162,172)(164,174)(166,176)(168,178)(170,180)(182,192)(184,194)(186,196)(188,198)(190,200)(201,211)(203,213)(205,215)(207,217)(209,219)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,47,85,193)(2,192,86,46)(3,45,87,191)(4,190,88,44)(5,43,89,189)(6,188,90,42)(7,41,91,187)(8,186,92,60)(9,59,93,185)(10,184,94,58)(11,57,95,183)(12,182,96,56)(13,55,97,181)(14,200,98,54)(15,53,99,199)(16,198,100,52)(17,51,81,197)(18,196,82,50)(19,49,83,195)(20,194,84,48)(21,172,123,113)(22,112,124,171)(23,170,125,111)(24,110,126,169)(25,168,127,109)(26,108,128,167)(27,166,129,107)(28,106,130,165)(29,164,131,105)(30,104,132,163)(31,162,133,103)(32,102,134,161)(33,180,135,101)(34,120,136,179)(35,178,137,119)(36,118,138,177)(37,176,139,117)(38,116,140,175)(39,174,121,115)(40,114,122,173)(61,227,201,160)(62,159,202,226)(63,225,203,158)(64,157,204,224)(65,223,205,156)(66,155,206,222)(67,221,207,154)(68,153,208,240)(69,239,209,152)(70,151,210,238)(71,237,211,150)(72,149,212,236)(73,235,213,148)(74,147,214,234)(75,233,215,146)(76,145,216,232)(77,231,217,144)(78,143,218,230)(79,229,219,142)(80,141,220,228)>;
G:=Group( (1,70,161,85,210,102)(2,103,211,86,162,71)(3,72,163,87,212,104)(4,105,213,88,164,73)(5,74,165,89,214,106)(6,107,215,90,166,75)(7,76,167,91,216,108)(8,109,217,92,168,77)(9,78,169,93,218,110)(10,111,219,94,170,79)(11,80,171,95,220,112)(12,113,201,96,172,61)(13,62,173,97,202,114)(14,115,203,98,174,63)(15,64,175,99,204,116)(16,117,205,100,176,65)(17,66,177,81,206,118)(18,119,207,82,178,67)(19,68,179,83,208,120)(20,101,209,84,180,69)(21,182,227,123,56,160)(22,141,57,124,228,183)(23,184,229,125,58,142)(24,143,59,126,230,185)(25,186,231,127,60,144)(26,145,41,128,232,187)(27,188,233,129,42,146)(28,147,43,130,234,189)(29,190,235,131,44,148)(30,149,45,132,236,191)(31,192,237,133,46,150)(32,151,47,134,238,193)(33,194,239,135,48,152)(34,153,49,136,240,195)(35,196,221,137,50,154)(36,155,51,138,222,197)(37,198,223,139,52,156)(38,157,53,140,224,199)(39,200,225,121,54,158)(40,159,55,122,226,181), (1,95)(2,12)(3,97)(4,14)(5,99)(6,16)(7,81)(8,18)(9,83)(10,20)(11,85)(13,87)(15,89)(17,91)(19,93)(21,31)(22,134)(23,33)(24,136)(25,35)(26,138)(27,37)(28,140)(29,39)(30,122)(32,124)(34,126)(36,128)(38,130)(40,132)(41,197)(42,52)(43,199)(44,54)(45,181)(46,56)(47,183)(48,58)(49,185)(50,60)(51,187)(53,189)(55,191)(57,193)(59,195)(61,71)(62,212)(63,73)(64,214)(65,75)(66,216)(67,77)(68,218)(69,79)(70,220)(72,202)(74,204)(76,206)(78,208)(80,210)(82,92)(84,94)(86,96)(88,98)(90,100)(101,111)(102,171)(103,113)(104,173)(105,115)(106,175)(107,117)(108,177)(109,119)(110,179)(112,161)(114,163)(116,165)(118,167)(120,169)(121,131)(123,133)(125,135)(127,137)(129,139)(141,238)(142,152)(143,240)(144,154)(145,222)(146,156)(147,224)(148,158)(149,226)(150,160)(151,228)(153,230)(155,232)(157,234)(159,236)(162,172)(164,174)(166,176)(168,178)(170,180)(182,192)(184,194)(186,196)(188,198)(190,200)(201,211)(203,213)(205,215)(207,217)(209,219)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,47,85,193)(2,192,86,46)(3,45,87,191)(4,190,88,44)(5,43,89,189)(6,188,90,42)(7,41,91,187)(8,186,92,60)(9,59,93,185)(10,184,94,58)(11,57,95,183)(12,182,96,56)(13,55,97,181)(14,200,98,54)(15,53,99,199)(16,198,100,52)(17,51,81,197)(18,196,82,50)(19,49,83,195)(20,194,84,48)(21,172,123,113)(22,112,124,171)(23,170,125,111)(24,110,126,169)(25,168,127,109)(26,108,128,167)(27,166,129,107)(28,106,130,165)(29,164,131,105)(30,104,132,163)(31,162,133,103)(32,102,134,161)(33,180,135,101)(34,120,136,179)(35,178,137,119)(36,118,138,177)(37,176,139,117)(38,116,140,175)(39,174,121,115)(40,114,122,173)(61,227,201,160)(62,159,202,226)(63,225,203,158)(64,157,204,224)(65,223,205,156)(66,155,206,222)(67,221,207,154)(68,153,208,240)(69,239,209,152)(70,151,210,238)(71,237,211,150)(72,149,212,236)(73,235,213,148)(74,147,214,234)(75,233,215,146)(76,145,216,232)(77,231,217,144)(78,143,218,230)(79,229,219,142)(80,141,220,228) );
G=PermutationGroup([[(1,70,161,85,210,102),(2,103,211,86,162,71),(3,72,163,87,212,104),(4,105,213,88,164,73),(5,74,165,89,214,106),(6,107,215,90,166,75),(7,76,167,91,216,108),(8,109,217,92,168,77),(9,78,169,93,218,110),(10,111,219,94,170,79),(11,80,171,95,220,112),(12,113,201,96,172,61),(13,62,173,97,202,114),(14,115,203,98,174,63),(15,64,175,99,204,116),(16,117,205,100,176,65),(17,66,177,81,206,118),(18,119,207,82,178,67),(19,68,179,83,208,120),(20,101,209,84,180,69),(21,182,227,123,56,160),(22,141,57,124,228,183),(23,184,229,125,58,142),(24,143,59,126,230,185),(25,186,231,127,60,144),(26,145,41,128,232,187),(27,188,233,129,42,146),(28,147,43,130,234,189),(29,190,235,131,44,148),(30,149,45,132,236,191),(31,192,237,133,46,150),(32,151,47,134,238,193),(33,194,239,135,48,152),(34,153,49,136,240,195),(35,196,221,137,50,154),(36,155,51,138,222,197),(37,198,223,139,52,156),(38,157,53,140,224,199),(39,200,225,121,54,158),(40,159,55,122,226,181)], [(1,95),(2,12),(3,97),(4,14),(5,99),(6,16),(7,81),(8,18),(9,83),(10,20),(11,85),(13,87),(15,89),(17,91),(19,93),(21,31),(22,134),(23,33),(24,136),(25,35),(26,138),(27,37),(28,140),(29,39),(30,122),(32,124),(34,126),(36,128),(38,130),(40,132),(41,197),(42,52),(43,199),(44,54),(45,181),(46,56),(47,183),(48,58),(49,185),(50,60),(51,187),(53,189),(55,191),(57,193),(59,195),(61,71),(62,212),(63,73),(64,214),(65,75),(66,216),(67,77),(68,218),(69,79),(70,220),(72,202),(74,204),(76,206),(78,208),(80,210),(82,92),(84,94),(86,96),(88,98),(90,100),(101,111),(102,171),(103,113),(104,173),(105,115),(106,175),(107,117),(108,177),(109,119),(110,179),(112,161),(114,163),(116,165),(118,167),(120,169),(121,131),(123,133),(125,135),(127,137),(129,139),(141,238),(142,152),(143,240),(144,154),(145,222),(146,156),(147,224),(148,158),(149,226),(150,160),(151,228),(153,230),(155,232),(157,234),(159,236),(162,172),(164,174),(166,176),(168,178),(170,180),(182,192),(184,194),(186,196),(188,198),(190,200),(201,211),(203,213),(205,215),(207,217),(209,219),(221,231),(223,233),(225,235),(227,237),(229,239)], [(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)], [(1,47,85,193),(2,192,86,46),(3,45,87,191),(4,190,88,44),(5,43,89,189),(6,188,90,42),(7,41,91,187),(8,186,92,60),(9,59,93,185),(10,184,94,58),(11,57,95,183),(12,182,96,56),(13,55,97,181),(14,200,98,54),(15,53,99,199),(16,198,100,52),(17,51,81,197),(18,196,82,50),(19,49,83,195),(20,194,84,48),(21,172,123,113),(22,112,124,171),(23,170,125,111),(24,110,126,169),(25,168,127,109),(26,108,128,167),(27,166,129,107),(28,106,130,165),(29,164,131,105),(30,104,132,163),(31,162,133,103),(32,102,134,161),(33,180,135,101),(34,120,136,179),(35,178,137,119),(36,118,138,177),(37,176,139,117),(38,116,140,175),(39,174,121,115),(40,114,122,173),(61,227,201,160),(62,159,202,226),(63,225,203,158),(64,157,204,224),(65,223,205,156),(66,155,206,222),(67,221,207,154),(68,153,208,240),(69,239,209,152),(70,151,210,238),(71,237,211,150),(72,149,212,236),(73,235,213,148),(74,147,214,234),(75,233,215,146),(76,145,216,232),(77,231,217,144),(78,143,218,230),(79,229,219,142),(80,141,220,228)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 2 | 12 | 12 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D20 | D4⋊2S3 | S3×D5 | D4⋊2D5 | C3⋊D20 | C2×S3×D5 | C30.C23 |
| kernel | C6.(C2×D20) | D10⋊Dic3 | C6.Dic10 | C5×C6.D4 | C6×C5⋊D4 | C22×Dic15 | C2×C5⋊D4 | C2×C30 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C6.(C2×D20) ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 56 |
| 0 | 0 | 0 | 0 | 56 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 25 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 41 | 0 | 0 | 0 | 0 | 0 |
| 49 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 46 | 0 | 0 |
| 0 | 0 | 37 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 55 |
| 0 | 0 | 0 | 0 | 8 | 13 |
| 2 | 47 | 0 | 0 | 0 | 0 |
| 22 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 7 |
| 0 | 0 | 0 | 0 | 4 | 41 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,39,56,0,0,0,0,56,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[41,49,0,0,0,0,0,3,0,0,0,0,0,0,35,37,0,0,0,0,46,26,0,0,0,0,0,0,48,8,0,0,0,0,55,13],[2,22,0,0,0,0,47,59,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,20,4,0,0,0,0,7,41] >;
C6.(C2×D20) in GAP, Magma, Sage, TeX
C_6.(C_2\times D_{20}) % in TeX
G:=Group("C6.(C2xD20)"); // GroupNames label
G:=SmallGroup(480,613);
// by ID
G=gap.SmallGroup(480,613);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations