metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic3⋊5D5, (C2×C20).222D6, (C2×C12).10D10, C6.Dic10⋊5C2, (C22×D5).6D6, C15⋊5(C42⋊2C2), (Dic3×Dic5)⋊9C2, D10⋊C4.7S3, C6.65(C4○D20), (C2×C30).51C23, C30.4Q8⋊17C2, (C2×Dic5).95D6, C5⋊4(C23.8D6), C30.106(C4○D4), C10.52(C4○D12), C6.21(D4⋊2D5), C10.6(D4⋊2S3), C2.9(D12⋊5D5), C2.9(D20⋊S3), (C2×C60).315C22, C6.25(Q8⋊2D5), (C2×Dic3).12D10, D10⋊Dic3.5C2, (C6×Dic5).29C22, C2.13(Dic5.D6), (C2×Dic15).53C22, (C10×Dic3).31C22, C3⋊4(C4⋊C4⋊D5), (C2×C4).34(S3×D5), (D5×C2×C6).4C22, (C5×C4⋊Dic3)⋊16C2, C22.138(C2×S3×D5), (C2×C6).63(C22×D5), (C2×C10).63(C22×S3), (C3×D10⋊C4).10C2, SmallGroup(480,437)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12).D10
G = < a,b,c,d | a2=b12=1, c10=a, d2=ab6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b6c9 >
Subgroups: 556 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C5×C4⋊C4, C23.8D6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C4⋊C4⋊D5, Dic3×Dic5, D10⋊Dic3, C6.Dic10, C3×D10⋊C4, C5×C4⋊Dic3, C30.4Q8, (C2×C12).D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, S3×D5, C4○D20, D4⋊2D5, Q8⋊2D5, C23.8D6, C2×S3×D5, C4⋊C4⋊D5, D20⋊S3, D12⋊5D5, Dic5.D6, (C2×C12).D10
►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 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(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)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(1 132 30 164 198 227 49 110 93 61 149 209)(2 210 150 62 94 111 50 228 199 165 31 133)(3 134 32 166 200 229 51 112 95 63 151 211)(4 212 152 64 96 113 52 230 181 167 33 135)(5 136 34 168 182 231 53 114 97 65 153 213)(6 214 154 66 98 115 54 232 183 169 35 137)(7 138 36 170 184 233 55 116 99 67 155 215)(8 216 156 68 100 117 56 234 185 171 37 139)(9 140 38 172 186 235 57 118 81 69 157 217)(10 218 158 70 82 119 58 236 187 173 39 121)(11 122 40 174 188 237 59 120 83 71 159 219)(12 220 160 72 84 101 60 238 189 175 21 123)(13 124 22 176 190 239 41 102 85 73 141 201)(14 202 142 74 86 103 42 240 191 177 23 125)(15 126 24 178 192 221 43 104 87 75 143 203)(16 204 144 76 88 105 44 222 193 179 25 127)(17 128 26 180 194 223 45 106 89 77 145 205)(18 206 146 78 90 107 46 224 195 161 27 129)(19 130 28 162 196 225 47 108 91 79 147 207)(20 208 148 80 92 109 48 226 197 163 29 131)
(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 10 59 48)(2 47 60 9)(3 8 41 46)(4 45 42 7)(5 6 43 44)(11 20 49 58)(12 57 50 19)(13 18 51 56)(14 55 52 17)(15 16 53 54)(21 81 94 28)(22 27 95 100)(23 99 96 26)(24 25 97 98)(29 93 82 40)(30 39 83 92)(31 91 84 38)(32 37 85 90)(33 89 86 36)(34 35 87 88)(61 80 174 173)(62 172 175 79)(63 78 176 171)(64 170 177 77)(65 76 178 169)(66 168 179 75)(67 74 180 167)(68 166 161 73)(69 72 162 165)(70 164 163 71)(101 130 133 118)(102 117 134 129)(103 128 135 116)(104 115 136 127)(105 126 137 114)(106 113 138 125)(107 124 139 112)(108 111 140 123)(109 122 121 110)(119 132 131 120)(141 146 200 185)(142 184 181 145)(143 144 182 183)(147 160 186 199)(148 198 187 159)(149 158 188 197)(150 196 189 157)(151 156 190 195)(152 194 191 155)(153 154 192 193)(201 216 229 224)(202 223 230 215)(203 214 231 222)(204 221 232 213)(205 212 233 240)(206 239 234 211)(207 210 235 238)(208 237 236 209)(217 220 225 228)(218 227 226 219)
G:=sub<Sym(240)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(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)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (1,132,30,164,198,227,49,110,93,61,149,209)(2,210,150,62,94,111,50,228,199,165,31,133)(3,134,32,166,200,229,51,112,95,63,151,211)(4,212,152,64,96,113,52,230,181,167,33,135)(5,136,34,168,182,231,53,114,97,65,153,213)(6,214,154,66,98,115,54,232,183,169,35,137)(7,138,36,170,184,233,55,116,99,67,155,215)(8,216,156,68,100,117,56,234,185,171,37,139)(9,140,38,172,186,235,57,118,81,69,157,217)(10,218,158,70,82,119,58,236,187,173,39,121)(11,122,40,174,188,237,59,120,83,71,159,219)(12,220,160,72,84,101,60,238,189,175,21,123)(13,124,22,176,190,239,41,102,85,73,141,201)(14,202,142,74,86,103,42,240,191,177,23,125)(15,126,24,178,192,221,43,104,87,75,143,203)(16,204,144,76,88,105,44,222,193,179,25,127)(17,128,26,180,194,223,45,106,89,77,145,205)(18,206,146,78,90,107,46,224,195,161,27,129)(19,130,28,162,196,225,47,108,91,79,147,207)(20,208,148,80,92,109,48,226,197,163,29,131), (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), (1,10,59,48)(2,47,60,9)(3,8,41,46)(4,45,42,7)(5,6,43,44)(11,20,49,58)(12,57,50,19)(13,18,51,56)(14,55,52,17)(15,16,53,54)(21,81,94,28)(22,27,95,100)(23,99,96,26)(24,25,97,98)(29,93,82,40)(30,39,83,92)(31,91,84,38)(32,37,85,90)(33,89,86,36)(34,35,87,88)(61,80,174,173)(62,172,175,79)(63,78,176,171)(64,170,177,77)(65,76,178,169)(66,168,179,75)(67,74,180,167)(68,166,161,73)(69,72,162,165)(70,164,163,71)(101,130,133,118)(102,117,134,129)(103,128,135,116)(104,115,136,127)(105,126,137,114)(106,113,138,125)(107,124,139,112)(108,111,140,123)(109,122,121,110)(119,132,131,120)(141,146,200,185)(142,184,181,145)(143,144,182,183)(147,160,186,199)(148,198,187,159)(149,158,188,197)(150,196,189,157)(151,156,190,195)(152,194,191,155)(153,154,192,193)(201,216,229,224)(202,223,230,215)(203,214,231,222)(204,221,232,213)(205,212,233,240)(206,239,234,211)(207,210,235,238)(208,237,236,209)(217,220,225,228)(218,227,226,219)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(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)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (1,132,30,164,198,227,49,110,93,61,149,209)(2,210,150,62,94,111,50,228,199,165,31,133)(3,134,32,166,200,229,51,112,95,63,151,211)(4,212,152,64,96,113,52,230,181,167,33,135)(5,136,34,168,182,231,53,114,97,65,153,213)(6,214,154,66,98,115,54,232,183,169,35,137)(7,138,36,170,184,233,55,116,99,67,155,215)(8,216,156,68,100,117,56,234,185,171,37,139)(9,140,38,172,186,235,57,118,81,69,157,217)(10,218,158,70,82,119,58,236,187,173,39,121)(11,122,40,174,188,237,59,120,83,71,159,219)(12,220,160,72,84,101,60,238,189,175,21,123)(13,124,22,176,190,239,41,102,85,73,141,201)(14,202,142,74,86,103,42,240,191,177,23,125)(15,126,24,178,192,221,43,104,87,75,143,203)(16,204,144,76,88,105,44,222,193,179,25,127)(17,128,26,180,194,223,45,106,89,77,145,205)(18,206,146,78,90,107,46,224,195,161,27,129)(19,130,28,162,196,225,47,108,91,79,147,207)(20,208,148,80,92,109,48,226,197,163,29,131), (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), (1,10,59,48)(2,47,60,9)(3,8,41,46)(4,45,42,7)(5,6,43,44)(11,20,49,58)(12,57,50,19)(13,18,51,56)(14,55,52,17)(15,16,53,54)(21,81,94,28)(22,27,95,100)(23,99,96,26)(24,25,97,98)(29,93,82,40)(30,39,83,92)(31,91,84,38)(32,37,85,90)(33,89,86,36)(34,35,87,88)(61,80,174,173)(62,172,175,79)(63,78,176,171)(64,170,177,77)(65,76,178,169)(66,168,179,75)(67,74,180,167)(68,166,161,73)(69,72,162,165)(70,164,163,71)(101,130,133,118)(102,117,134,129)(103,128,135,116)(104,115,136,127)(105,126,137,114)(106,113,138,125)(107,124,139,112)(108,111,140,123)(109,122,121,110)(119,132,131,120)(141,146,200,185)(142,184,181,145)(143,144,182,183)(147,160,186,199)(148,198,187,159)(149,158,188,197)(150,196,189,157)(151,156,190,195)(152,194,191,155)(153,154,192,193)(201,216,229,224)(202,223,230,215)(203,214,231,222)(204,221,232,213)(205,212,233,240)(206,239,234,211)(207,210,235,238)(208,237,236,209)(217,220,225,228)(218,227,226,219) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(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),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178),(169,179),(170,180),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199),(190,200),(201,211),(202,212),(203,213),(204,214),(205,215),(206,216),(207,217),(208,218),(209,219),(210,220),(221,231),(222,232),(223,233),(224,234),(225,235),(226,236),(227,237),(228,238),(229,239),(230,240)], [(1,132,30,164,198,227,49,110,93,61,149,209),(2,210,150,62,94,111,50,228,199,165,31,133),(3,134,32,166,200,229,51,112,95,63,151,211),(4,212,152,64,96,113,52,230,181,167,33,135),(5,136,34,168,182,231,53,114,97,65,153,213),(6,214,154,66,98,115,54,232,183,169,35,137),(7,138,36,170,184,233,55,116,99,67,155,215),(8,216,156,68,100,117,56,234,185,171,37,139),(9,140,38,172,186,235,57,118,81,69,157,217),(10,218,158,70,82,119,58,236,187,173,39,121),(11,122,40,174,188,237,59,120,83,71,159,219),(12,220,160,72,84,101,60,238,189,175,21,123),(13,124,22,176,190,239,41,102,85,73,141,201),(14,202,142,74,86,103,42,240,191,177,23,125),(15,126,24,178,192,221,43,104,87,75,143,203),(16,204,144,76,88,105,44,222,193,179,25,127),(17,128,26,180,194,223,45,106,89,77,145,205),(18,206,146,78,90,107,46,224,195,161,27,129),(19,130,28,162,196,225,47,108,91,79,147,207),(20,208,148,80,92,109,48,226,197,163,29,131)], [(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)], [(1,10,59,48),(2,47,60,9),(3,8,41,46),(4,45,42,7),(5,6,43,44),(11,20,49,58),(12,57,50,19),(13,18,51,56),(14,55,52,17),(15,16,53,54),(21,81,94,28),(22,27,95,100),(23,99,96,26),(24,25,97,98),(29,93,82,40),(30,39,83,92),(31,91,84,38),(32,37,85,90),(33,89,86,36),(34,35,87,88),(61,80,174,173),(62,172,175,79),(63,78,176,171),(64,170,177,77),(65,76,178,169),(66,168,179,75),(67,74,180,167),(68,166,161,73),(69,72,162,165),(70,164,163,71),(101,130,133,118),(102,117,134,129),(103,128,135,116),(104,115,136,127),(105,126,137,114),(106,113,138,125),(107,124,139,112),(108,111,140,123),(109,122,121,110),(119,132,131,120),(141,146,200,185),(142,184,181,145),(143,144,182,183),(147,160,186,199),(148,198,187,159),(149,158,188,197),(150,196,189,157),(151,156,190,195),(152,194,191,155),(153,154,192,193),(201,216,229,224),(202,223,230,215),(203,214,231,222),(204,221,232,213),(205,212,233,240),(206,239,234,211),(207,210,235,238),(208,237,236,209),(217,220,225,228),(218,227,226,219)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊S3 | D12⋊5D5 | Dic5.D6 |
| kernel | (C2×C12).D10 | Dic3×Dic5 | D10⋊Dic3 | C6.Dic10 | C3×D10⋊C4 | C5×C4⋊Dic3 | C30.4Q8 | D10⋊C4 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C10 | C6 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 6 | 4 | 2 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C2×C12).D10 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 40 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 16 | 0 | 0 |
| 0 | 0 | 45 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 |
| 0 | 0 | 0 | 0 | 12 | 60 |
| 60 | 13 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 4 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 50 |
| 0 | 0 | 0 | 0 | 5 | 42 |
| 1 | 48 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 50 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 50 |
| 0 | 0 | 0 | 0 | 5 | 42 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,40,50,0,0,0,0,0,0,47,45,0,0,0,0,16,14,0,0,0,0,0,0,2,12,0,0,0,0,15,60],[60,28,0,0,0,0,13,1,0,0,0,0,0,0,57,4,0,0,0,0,57,50,0,0,0,0,0,0,19,5,0,0,0,0,50,42],[1,0,0,0,0,0,48,60,0,0,0,0,0,0,57,50,0,0,0,0,57,4,0,0,0,0,0,0,19,5,0,0,0,0,50,42] >;
(C2×C12).D10 in GAP, Magma, Sage, TeX
(C_2\times C_{12}).D_{10} % in TeX
G:=Group("(C2xC12).D10"); // GroupNames label
G:=SmallGroup(480,437);
// by ID
G=gap.SmallGroup(480,437);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=1,c^10=a,d^2=a*b^6,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=b^6*c^9>;
// generators/relations