G = C10.(C2×D12) order 480 = 25·3·5
21st non-split extension by C10 of C2×D12 acting via C2×D12/C22×S3=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30).73D4, C23.D5⋊5S3, C10.67(C2×D12), (C2×C10).47D12, C30.222(C2×D4), D30⋊4C4⋊26C2, C23.28(S3×D5), C6.81(C4○D20), C30.Q8⋊31C2, (C2×Dic5).57D6, (C22×Dic3)⋊4D5, (C22×C6).27D10, C30.140(C4○D4), (C2×C30).184C23, (C22×C10).104D6, C5⋊4(C23.21D6), C22.6(C5⋊D12), C10.53(D4⋊2S3), (C2×Dic3).163D10, C3⋊3(C23.23D10), C15⋊19(C22.D4), (C22×C30).46C22, C2.26(Dic5.D6), (C6×Dic5).107C22, (C22×D15).60C22, (C2×Dic15).128C22, (C10×Dic3).199C22, (Dic3×C2×C10)⋊4C2, C6.21(C2×C5⋊D4), (C2×C15⋊7D4).9C2, C2.22(C2×C5⋊D12), (C3×C23.D5)⋊5C2, C22.224(C2×S3×D5), (C2×C6).17(C5⋊D4), (C2×C6).196(C22×D5), (C2×C10).196(C22×S3), SmallGroup(480,618)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.(C2×D12)
G = < a,b,c,d | a10=b2=c12=1, d2=a5, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 812 in 156 conjugacy classes, 52 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, 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, C22×S3, C22×C6, D15, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, D30, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.21D6, C6×Dic5, C10×Dic3, C10×Dic3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, C23.23D10, D30⋊4C4, C30.Q8, C3×C23.D5, Dic3×C2×C10, C2×C15⋊7D4, C10.(C2×D12)
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C22.D4, C5⋊D4, C22×D5, C2×D12, D4⋊2S3, S3×D5, C4○D20, C2×C5⋊D4, C23.21D6, C5⋊D12, C2×S3×D5, C23.23D10, Dic5.D6, C2×C5⋊D12, C10.(C2×D12)
►On 240 points
Generators in S240
(1 113 18 186 137 81 175 239 90 211)(2 212 91 240 176 82 138 187 19 114)(3 115 20 188 139 83 177 229 92 213)(4 214 93 230 178 84 140 189 21 116)(5 117 22 190 141 73 179 231 94 215)(6 216 95 232 180 74 142 191 23 118)(7 119 24 192 143 75 169 233 96 205)(8 206 85 234 170 76 144 181 13 120)(9 109 14 182 133 77 171 235 86 207)(10 208 87 236 172 78 134 183 15 110)(11 111 16 184 135 79 173 237 88 209)(12 210 89 238 174 80 136 185 17 112)(25 126 147 163 200 224 104 58 61 39)(26 40 62 59 105 225 201 164 148 127)(27 128 149 165 202 226 106 60 63 41)(28 42 64 49 107 227 203 166 150 129)(29 130 151 167 204 228 108 50 65 43)(30 44 66 51 97 217 193 168 152 131)(31 132 153 157 194 218 98 52 67 45)(32 46 68 53 99 219 195 158 154 121)(33 122 155 159 196 220 100 54 69 47)(34 48 70 55 101 221 197 160 156 123)(35 124 145 161 198 222 102 56 71 37)(36 38 72 57 103 223 199 162 146 125)
(1 7)(2 76)(3 9)(4 78)(5 11)(6 80)(8 82)(10 84)(12 74)(13 240)(14 20)(15 230)(16 22)(17 232)(18 24)(19 234)(21 236)(23 238)(25 218)(26 32)(27 220)(28 34)(29 222)(30 36)(31 224)(33 226)(35 228)(37 204)(38 44)(39 194)(40 46)(41 196)(42 48)(43 198)(45 200)(47 202)(49 55)(50 145)(51 57)(52 147)(53 59)(54 149)(56 151)(58 153)(60 155)(61 157)(62 68)(63 159)(64 70)(65 161)(66 72)(67 163)(69 165)(71 167)(73 79)(75 81)(77 83)(85 187)(86 92)(87 189)(88 94)(89 191)(90 96)(91 181)(93 183)(95 185)(97 103)(98 126)(99 105)(100 128)(101 107)(102 130)(104 132)(106 122)(108 124)(109 115)(110 178)(111 117)(112 180)(113 119)(114 170)(116 172)(118 174)(120 176)(121 127)(123 129)(125 131)(133 139)(134 214)(135 141)(136 216)(137 143)(138 206)(140 208)(142 210)(144 212)(146 152)(148 154)(150 156)(158 164)(160 166)(162 168)(169 175)(171 177)(173 179)(182 188)(184 190)(186 192)(193 199)(195 201)(197 203)(205 211)(207 213)(209 215)(217 223)(219 225)(221 227)(229 235)(231 237)(233 239)
(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)
(1 228 81 29)(2 227 82 28)(3 226 83 27)(4 225 84 26)(5 224 73 25)(6 223 74 36)(7 222 75 35)(8 221 76 34)(9 220 77 33)(10 219 78 32)(11 218 79 31)(12 217 80 30)(13 160 234 70)(14 159 235 69)(15 158 236 68)(16 157 237 67)(17 168 238 66)(18 167 239 65)(19 166 240 64)(20 165 229 63)(21 164 230 62)(22 163 231 61)(23 162 232 72)(24 161 233 71)(37 119 198 169)(38 118 199 180)(39 117 200 179)(40 116 201 178)(41 115 202 177)(42 114 203 176)(43 113 204 175)(44 112 193 174)(45 111 194 173)(46 110 195 172)(47 109 196 171)(48 120 197 170)(49 187 150 91)(50 186 151 90)(51 185 152 89)(52 184 153 88)(53 183 154 87)(54 182 155 86)(55 181 156 85)(56 192 145 96)(57 191 146 95)(58 190 147 94)(59 189 148 93)(60 188 149 92)(97 136 131 210)(98 135 132 209)(99 134 121 208)(100 133 122 207)(101 144 123 206)(102 143 124 205)(103 142 125 216)(104 141 126 215)(105 140 127 214)(106 139 128 213)(107 138 129 212)(108 137 130 211)
G:=sub<Sym(240)| (1,113,18,186,137,81,175,239,90,211)(2,212,91,240,176,82,138,187,19,114)(3,115,20,188,139,83,177,229,92,213)(4,214,93,230,178,84,140,189,21,116)(5,117,22,190,141,73,179,231,94,215)(6,216,95,232,180,74,142,191,23,118)(7,119,24,192,143,75,169,233,96,205)(8,206,85,234,170,76,144,181,13,120)(9,109,14,182,133,77,171,235,86,207)(10,208,87,236,172,78,134,183,15,110)(11,111,16,184,135,79,173,237,88,209)(12,210,89,238,174,80,136,185,17,112)(25,126,147,163,200,224,104,58,61,39)(26,40,62,59,105,225,201,164,148,127)(27,128,149,165,202,226,106,60,63,41)(28,42,64,49,107,227,203,166,150,129)(29,130,151,167,204,228,108,50,65,43)(30,44,66,51,97,217,193,168,152,131)(31,132,153,157,194,218,98,52,67,45)(32,46,68,53,99,219,195,158,154,121)(33,122,155,159,196,220,100,54,69,47)(34,48,70,55,101,221,197,160,156,123)(35,124,145,161,198,222,102,56,71,37)(36,38,72,57,103,223,199,162,146,125), (1,7)(2,76)(3,9)(4,78)(5,11)(6,80)(8,82)(10,84)(12,74)(13,240)(14,20)(15,230)(16,22)(17,232)(18,24)(19,234)(21,236)(23,238)(25,218)(26,32)(27,220)(28,34)(29,222)(30,36)(31,224)(33,226)(35,228)(37,204)(38,44)(39,194)(40,46)(41,196)(42,48)(43,198)(45,200)(47,202)(49,55)(50,145)(51,57)(52,147)(53,59)(54,149)(56,151)(58,153)(60,155)(61,157)(62,68)(63,159)(64,70)(65,161)(66,72)(67,163)(69,165)(71,167)(73,79)(75,81)(77,83)(85,187)(86,92)(87,189)(88,94)(89,191)(90,96)(91,181)(93,183)(95,185)(97,103)(98,126)(99,105)(100,128)(101,107)(102,130)(104,132)(106,122)(108,124)(109,115)(110,178)(111,117)(112,180)(113,119)(114,170)(116,172)(118,174)(120,176)(121,127)(123,129)(125,131)(133,139)(134,214)(135,141)(136,216)(137,143)(138,206)(140,208)(142,210)(144,212)(146,152)(148,154)(150,156)(158,164)(160,166)(162,168)(169,175)(171,177)(173,179)(182,188)(184,190)(186,192)(193,199)(195,201)(197,203)(205,211)(207,213)(209,215)(217,223)(219,225)(221,227)(229,235)(231,237)(233,239), (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), (1,228,81,29)(2,227,82,28)(3,226,83,27)(4,225,84,26)(5,224,73,25)(6,223,74,36)(7,222,75,35)(8,221,76,34)(9,220,77,33)(10,219,78,32)(11,218,79,31)(12,217,80,30)(13,160,234,70)(14,159,235,69)(15,158,236,68)(16,157,237,67)(17,168,238,66)(18,167,239,65)(19,166,240,64)(20,165,229,63)(21,164,230,62)(22,163,231,61)(23,162,232,72)(24,161,233,71)(37,119,198,169)(38,118,199,180)(39,117,200,179)(40,116,201,178)(41,115,202,177)(42,114,203,176)(43,113,204,175)(44,112,193,174)(45,111,194,173)(46,110,195,172)(47,109,196,171)(48,120,197,170)(49,187,150,91)(50,186,151,90)(51,185,152,89)(52,184,153,88)(53,183,154,87)(54,182,155,86)(55,181,156,85)(56,192,145,96)(57,191,146,95)(58,190,147,94)(59,189,148,93)(60,188,149,92)(97,136,131,210)(98,135,132,209)(99,134,121,208)(100,133,122,207)(101,144,123,206)(102,143,124,205)(103,142,125,216)(104,141,126,215)(105,140,127,214)(106,139,128,213)(107,138,129,212)(108,137,130,211)>;
G:=Group( (1,113,18,186,137,81,175,239,90,211)(2,212,91,240,176,82,138,187,19,114)(3,115,20,188,139,83,177,229,92,213)(4,214,93,230,178,84,140,189,21,116)(5,117,22,190,141,73,179,231,94,215)(6,216,95,232,180,74,142,191,23,118)(7,119,24,192,143,75,169,233,96,205)(8,206,85,234,170,76,144,181,13,120)(9,109,14,182,133,77,171,235,86,207)(10,208,87,236,172,78,134,183,15,110)(11,111,16,184,135,79,173,237,88,209)(12,210,89,238,174,80,136,185,17,112)(25,126,147,163,200,224,104,58,61,39)(26,40,62,59,105,225,201,164,148,127)(27,128,149,165,202,226,106,60,63,41)(28,42,64,49,107,227,203,166,150,129)(29,130,151,167,204,228,108,50,65,43)(30,44,66,51,97,217,193,168,152,131)(31,132,153,157,194,218,98,52,67,45)(32,46,68,53,99,219,195,158,154,121)(33,122,155,159,196,220,100,54,69,47)(34,48,70,55,101,221,197,160,156,123)(35,124,145,161,198,222,102,56,71,37)(36,38,72,57,103,223,199,162,146,125), (1,7)(2,76)(3,9)(4,78)(5,11)(6,80)(8,82)(10,84)(12,74)(13,240)(14,20)(15,230)(16,22)(17,232)(18,24)(19,234)(21,236)(23,238)(25,218)(26,32)(27,220)(28,34)(29,222)(30,36)(31,224)(33,226)(35,228)(37,204)(38,44)(39,194)(40,46)(41,196)(42,48)(43,198)(45,200)(47,202)(49,55)(50,145)(51,57)(52,147)(53,59)(54,149)(56,151)(58,153)(60,155)(61,157)(62,68)(63,159)(64,70)(65,161)(66,72)(67,163)(69,165)(71,167)(73,79)(75,81)(77,83)(85,187)(86,92)(87,189)(88,94)(89,191)(90,96)(91,181)(93,183)(95,185)(97,103)(98,126)(99,105)(100,128)(101,107)(102,130)(104,132)(106,122)(108,124)(109,115)(110,178)(111,117)(112,180)(113,119)(114,170)(116,172)(118,174)(120,176)(121,127)(123,129)(125,131)(133,139)(134,214)(135,141)(136,216)(137,143)(138,206)(140,208)(142,210)(144,212)(146,152)(148,154)(150,156)(158,164)(160,166)(162,168)(169,175)(171,177)(173,179)(182,188)(184,190)(186,192)(193,199)(195,201)(197,203)(205,211)(207,213)(209,215)(217,223)(219,225)(221,227)(229,235)(231,237)(233,239), (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), (1,228,81,29)(2,227,82,28)(3,226,83,27)(4,225,84,26)(5,224,73,25)(6,223,74,36)(7,222,75,35)(8,221,76,34)(9,220,77,33)(10,219,78,32)(11,218,79,31)(12,217,80,30)(13,160,234,70)(14,159,235,69)(15,158,236,68)(16,157,237,67)(17,168,238,66)(18,167,239,65)(19,166,240,64)(20,165,229,63)(21,164,230,62)(22,163,231,61)(23,162,232,72)(24,161,233,71)(37,119,198,169)(38,118,199,180)(39,117,200,179)(40,116,201,178)(41,115,202,177)(42,114,203,176)(43,113,204,175)(44,112,193,174)(45,111,194,173)(46,110,195,172)(47,109,196,171)(48,120,197,170)(49,187,150,91)(50,186,151,90)(51,185,152,89)(52,184,153,88)(53,183,154,87)(54,182,155,86)(55,181,156,85)(56,192,145,96)(57,191,146,95)(58,190,147,94)(59,189,148,93)(60,188,149,92)(97,136,131,210)(98,135,132,209)(99,134,121,208)(100,133,122,207)(101,144,123,206)(102,143,124,205)(103,142,125,216)(104,141,126,215)(105,140,127,214)(106,139,128,213)(107,138,129,212)(108,137,130,211) );
G=PermutationGroup([[(1,113,18,186,137,81,175,239,90,211),(2,212,91,240,176,82,138,187,19,114),(3,115,20,188,139,83,177,229,92,213),(4,214,93,230,178,84,140,189,21,116),(5,117,22,190,141,73,179,231,94,215),(6,216,95,232,180,74,142,191,23,118),(7,119,24,192,143,75,169,233,96,205),(8,206,85,234,170,76,144,181,13,120),(9,109,14,182,133,77,171,235,86,207),(10,208,87,236,172,78,134,183,15,110),(11,111,16,184,135,79,173,237,88,209),(12,210,89,238,174,80,136,185,17,112),(25,126,147,163,200,224,104,58,61,39),(26,40,62,59,105,225,201,164,148,127),(27,128,149,165,202,226,106,60,63,41),(28,42,64,49,107,227,203,166,150,129),(29,130,151,167,204,228,108,50,65,43),(30,44,66,51,97,217,193,168,152,131),(31,132,153,157,194,218,98,52,67,45),(32,46,68,53,99,219,195,158,154,121),(33,122,155,159,196,220,100,54,69,47),(34,48,70,55,101,221,197,160,156,123),(35,124,145,161,198,222,102,56,71,37),(36,38,72,57,103,223,199,162,146,125)], [(1,7),(2,76),(3,9),(4,78),(5,11),(6,80),(8,82),(10,84),(12,74),(13,240),(14,20),(15,230),(16,22),(17,232),(18,24),(19,234),(21,236),(23,238),(25,218),(26,32),(27,220),(28,34),(29,222),(30,36),(31,224),(33,226),(35,228),(37,204),(38,44),(39,194),(40,46),(41,196),(42,48),(43,198),(45,200),(47,202),(49,55),(50,145),(51,57),(52,147),(53,59),(54,149),(56,151),(58,153),(60,155),(61,157),(62,68),(63,159),(64,70),(65,161),(66,72),(67,163),(69,165),(71,167),(73,79),(75,81),(77,83),(85,187),(86,92),(87,189),(88,94),(89,191),(90,96),(91,181),(93,183),(95,185),(97,103),(98,126),(99,105),(100,128),(101,107),(102,130),(104,132),(106,122),(108,124),(109,115),(110,178),(111,117),(112,180),(113,119),(114,170),(116,172),(118,174),(120,176),(121,127),(123,129),(125,131),(133,139),(134,214),(135,141),(136,216),(137,143),(138,206),(140,208),(142,210),(144,212),(146,152),(148,154),(150,156),(158,164),(160,166),(162,168),(169,175),(171,177),(173,179),(182,188),(184,190),(186,192),(193,199),(195,201),(197,203),(205,211),(207,213),(209,215),(217,223),(219,225),(221,227),(229,235),(231,237),(233,239)], [(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)], [(1,228,81,29),(2,227,82,28),(3,226,83,27),(4,225,84,26),(5,224,73,25),(6,223,74,36),(7,222,75,35),(8,221,76,34),(9,220,77,33),(10,219,78,32),(11,218,79,31),(12,217,80,30),(13,160,234,70),(14,159,235,69),(15,158,236,68),(16,157,237,67),(17,168,238,66),(18,167,239,65),(19,166,240,64),(20,165,229,63),(21,164,230,62),(22,163,231,61),(23,162,232,72),(24,161,233,71),(37,119,198,169),(38,118,199,180),(39,117,200,179),(40,116,201,178),(41,115,202,177),(42,114,203,176),(43,113,204,175),(44,112,193,174),(45,111,194,173),(46,110,195,172),(47,109,196,171),(48,120,197,170),(49,187,150,91),(50,186,151,90),(51,185,152,89),(52,184,153,88),(53,183,154,87),(54,182,155,86),(55,181,156,85),(56,192,145,96),(57,191,146,95),(58,190,147,94),(59,189,148,93),(60,188,149,92),(97,136,131,210),(98,135,132,209),(99,134,121,208),(100,133,122,207),(101,144,123,206),(102,143,124,205),(103,142,125,216),(104,141,126,215),(105,140,127,214),(106,139,128,213),(107,138,129,212),(108,137,130,211)]])
72 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 | 10A | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20P | 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 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 60 | 2 | 6 | 6 | 6 | 6 | 20 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D12 | C5⋊D4 | C4○D20 | D4⋊2S3 | S3×D5 | C5⋊D12 | C2×S3×D5 | Dic5.D6 |
| kernel | C10.(C2×D12) | D30⋊4C4 | C30.Q8 | C3×C23.D5 | Dic3×C2×C10 | C2×C15⋊7D4 | C23.D5 | C2×C30 | C22×Dic3 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C6 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C10.(C2×D12) ►in GL6(𝔽61)
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 41 | 41 |
| 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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 8 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 24 | 0 | 0 |
| 0 | 0 | 34 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 27 |
| 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,41,0,0,0,0,0,41],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,15,0,0,0,0,0,1],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,51,34,0,0,0,0,24,10,0,0,0,0,0,0,50,0,0,0,0,0,27,11] >;
C10.(C2×D12) in GAP, Magma, Sage, TeX
C_{10}.(C_2\times D_{12}) % in TeX
G:=Group("C10.(C2xD12)"); // GroupNames label
G:=SmallGroup(480,618);
// by ID
G=gap.SmallGroup(480,618);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^5,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations