direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.F5, C30⋊3M4(2), (C2×C60).4C4, C6⋊2(C4.F5), C60.51(C2×C4), (C4×D5).90D6, (D5×C12).7C4, C12.51(C2×F5), (C2×C12).10F5, C15⋊11(C2×M4(2)), (C4×D5).4Dic3, (C2×C20).6Dic3, C15⋊C8⋊11C22, C6.33(C22×F5), C30.71(C22×C4), C10⋊1(C4.Dic3), C20.12(C2×Dic3), (C2×Dic5).207D6, D10.13(C2×Dic3), (C22×D5).8Dic3, C10.2(C22×Dic3), Dic5.15(C2×Dic3), (D5×C12).115C22, (C3×Dic5).63C23, Dic5.49(C22×S3), (C6×Dic5).266C22, C3⋊3(C2×C4.F5), C4.12(C2×C3⋊F5), (D5×C2×C6).14C4, (C2×C4×D5).14S3, (C2×C4).7(C3⋊F5), C5⋊1(C2×C4.Dic3), C2.4(C22×C3⋊F5), (D5×C2×C12).17C2, (C2×C15⋊C8)⋊10C2, (C2×C6).44(C2×F5), (C2×C30).38(C2×C4), C22.17(C2×C3⋊F5), (C6×D5).57(C2×C4), (C3×Dic5).65(C2×C4), (C2×C10).14(C2×Dic3), SmallGroup(480,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.F5
G = < a,b,c,d | a2=b12=c5=1, d4=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 524 in 136 conjugacy classes, 65 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C4.Dic3, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4.Dic3, C15⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C2×C4.F5, C12.F5, C2×C15⋊C8, D5×C2×C12, C2×C12.F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, F5, C2×Dic3, C22×S3, C2×M4(2), C2×F5, C4.Dic3, C22×Dic3, C3⋊F5, C4.F5, C22×F5, C2×C4.Dic3, C2×C3⋊F5, C2×C4.F5, C12.F5, C22×C3⋊F5, C2×C12.F5
►On 240 points
Generators in S240
(1 80)(2 81)(3 82)(4 83)(5 84)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 121)(22 122)(23 123)(24 124)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 161)(38 162)(39 163)(40 164)(41 165)(42 166)(43 167)(44 168)(45 157)(46 158)(47 159)(48 160)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(85 178)(86 179)(87 180)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 201)(98 202)(99 203)(100 204)(101 193)(102 194)(103 195)(104 196)(105 197)(106 198)(107 199)(108 200)(109 153)(110 154)(111 155)(112 156)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 181)(144 182)(217 234)(218 235)(219 236)(220 237)(221 238)(222 239)(223 240)(224 229)(225 230)(226 231)(227 232)(228 233)
(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 71 174 181 203)(2 72 175 182 204)(3 61 176 183 193)(4 62 177 184 194)(5 63 178 185 195)(6 64 179 186 196)(7 65 180 187 197)(8 66 169 188 198)(9 67 170 189 199)(10 68 171 190 200)(11 69 172 191 201)(12 70 173 192 202)(13 236 57 153 48)(14 237 58 154 37)(15 238 59 155 38)(16 239 60 156 39)(17 240 49 145 40)(18 229 50 146 41)(19 230 51 147 42)(20 231 52 148 43)(21 232 53 149 44)(22 233 54 150 45)(23 234 55 151 46)(24 235 56 152 47)(25 116 167 132 226)(26 117 168 121 227)(27 118 157 122 228)(28 119 158 123 217)(29 120 159 124 218)(30 109 160 125 219)(31 110 161 126 220)(32 111 162 127 221)(33 112 163 128 222)(34 113 164 129 223)(35 114 165 130 224)(36 115 166 131 225)(73 214 86 136 104)(74 215 87 137 105)(75 216 88 138 106)(76 205 89 139 107)(77 206 90 140 108)(78 207 91 141 97)(79 208 92 142 98)(80 209 93 143 99)(81 210 94 144 100)(82 211 95 133 101)(83 212 96 134 102)(84 213 85 135 103)
(1 17 77 132 7 23 83 126)(2 16 78 131 8 22 84 125)(3 15 79 130 9 21 73 124)(4 14 80 129 10 20 74 123)(5 13 81 128 11 19 75 122)(6 24 82 127 12 18 76 121)(25 197 151 212 31 203 145 206)(26 196 152 211 32 202 146 205)(27 195 153 210 33 201 147 216)(28 194 154 209 34 200 148 215)(29 193 155 208 35 199 149 214)(30 204 156 207 36 198 150 213)(37 143 223 171 43 137 217 177)(38 142 224 170 44 136 218 176)(39 141 225 169 45 135 219 175)(40 140 226 180 46 134 220 174)(41 139 227 179 47 133 221 173)(42 138 228 178 48 144 222 172)(49 108 116 65 55 102 110 71)(50 107 117 64 56 101 111 70)(51 106 118 63 57 100 112 69)(52 105 119 62 58 99 113 68)(53 104 120 61 59 98 114 67)(54 103 109 72 60 97 115 66)(85 160 182 239 91 166 188 233)(86 159 183 238 92 165 189 232)(87 158 184 237 93 164 190 231)(88 157 185 236 94 163 191 230)(89 168 186 235 95 162 192 229)(90 167 187 234 96 161 181 240)
G:=sub<Sym(240)| (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,121)(22,122)(23,123)(24,124)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,157)(46,158)(47,159)(48,160)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(85,178)(86,179)(87,180)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,201)(98,202)(99,203)(100,204)(101,193)(102,194)(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,153)(110,154)(111,155)(112,156)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,181)(144,182)(217,234)(218,235)(219,236)(220,237)(221,238)(222,239)(223,240)(224,229)(225,230)(226,231)(227,232)(228,233), (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), (1,71,174,181,203)(2,72,175,182,204)(3,61,176,183,193)(4,62,177,184,194)(5,63,178,185,195)(6,64,179,186,196)(7,65,180,187,197)(8,66,169,188,198)(9,67,170,189,199)(10,68,171,190,200)(11,69,172,191,201)(12,70,173,192,202)(13,236,57,153,48)(14,237,58,154,37)(15,238,59,155,38)(16,239,60,156,39)(17,240,49,145,40)(18,229,50,146,41)(19,230,51,147,42)(20,231,52,148,43)(21,232,53,149,44)(22,233,54,150,45)(23,234,55,151,46)(24,235,56,152,47)(25,116,167,132,226)(26,117,168,121,227)(27,118,157,122,228)(28,119,158,123,217)(29,120,159,124,218)(30,109,160,125,219)(31,110,161,126,220)(32,111,162,127,221)(33,112,163,128,222)(34,113,164,129,223)(35,114,165,130,224)(36,115,166,131,225)(73,214,86,136,104)(74,215,87,137,105)(75,216,88,138,106)(76,205,89,139,107)(77,206,90,140,108)(78,207,91,141,97)(79,208,92,142,98)(80,209,93,143,99)(81,210,94,144,100)(82,211,95,133,101)(83,212,96,134,102)(84,213,85,135,103), (1,17,77,132,7,23,83,126)(2,16,78,131,8,22,84,125)(3,15,79,130,9,21,73,124)(4,14,80,129,10,20,74,123)(5,13,81,128,11,19,75,122)(6,24,82,127,12,18,76,121)(25,197,151,212,31,203,145,206)(26,196,152,211,32,202,146,205)(27,195,153,210,33,201,147,216)(28,194,154,209,34,200,148,215)(29,193,155,208,35,199,149,214)(30,204,156,207,36,198,150,213)(37,143,223,171,43,137,217,177)(38,142,224,170,44,136,218,176)(39,141,225,169,45,135,219,175)(40,140,226,180,46,134,220,174)(41,139,227,179,47,133,221,173)(42,138,228,178,48,144,222,172)(49,108,116,65,55,102,110,71)(50,107,117,64,56,101,111,70)(51,106,118,63,57,100,112,69)(52,105,119,62,58,99,113,68)(53,104,120,61,59,98,114,67)(54,103,109,72,60,97,115,66)(85,160,182,239,91,166,188,233)(86,159,183,238,92,165,189,232)(87,158,184,237,93,164,190,231)(88,157,185,236,94,163,191,230)(89,168,186,235,95,162,192,229)(90,167,187,234,96,161,181,240)>;
G:=Group( (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,121)(22,122)(23,123)(24,124)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,157)(46,158)(47,159)(48,160)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(85,178)(86,179)(87,180)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,201)(98,202)(99,203)(100,204)(101,193)(102,194)(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,153)(110,154)(111,155)(112,156)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,181)(144,182)(217,234)(218,235)(219,236)(220,237)(221,238)(222,239)(223,240)(224,229)(225,230)(226,231)(227,232)(228,233), (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), (1,71,174,181,203)(2,72,175,182,204)(3,61,176,183,193)(4,62,177,184,194)(5,63,178,185,195)(6,64,179,186,196)(7,65,180,187,197)(8,66,169,188,198)(9,67,170,189,199)(10,68,171,190,200)(11,69,172,191,201)(12,70,173,192,202)(13,236,57,153,48)(14,237,58,154,37)(15,238,59,155,38)(16,239,60,156,39)(17,240,49,145,40)(18,229,50,146,41)(19,230,51,147,42)(20,231,52,148,43)(21,232,53,149,44)(22,233,54,150,45)(23,234,55,151,46)(24,235,56,152,47)(25,116,167,132,226)(26,117,168,121,227)(27,118,157,122,228)(28,119,158,123,217)(29,120,159,124,218)(30,109,160,125,219)(31,110,161,126,220)(32,111,162,127,221)(33,112,163,128,222)(34,113,164,129,223)(35,114,165,130,224)(36,115,166,131,225)(73,214,86,136,104)(74,215,87,137,105)(75,216,88,138,106)(76,205,89,139,107)(77,206,90,140,108)(78,207,91,141,97)(79,208,92,142,98)(80,209,93,143,99)(81,210,94,144,100)(82,211,95,133,101)(83,212,96,134,102)(84,213,85,135,103), (1,17,77,132,7,23,83,126)(2,16,78,131,8,22,84,125)(3,15,79,130,9,21,73,124)(4,14,80,129,10,20,74,123)(5,13,81,128,11,19,75,122)(6,24,82,127,12,18,76,121)(25,197,151,212,31,203,145,206)(26,196,152,211,32,202,146,205)(27,195,153,210,33,201,147,216)(28,194,154,209,34,200,148,215)(29,193,155,208,35,199,149,214)(30,204,156,207,36,198,150,213)(37,143,223,171,43,137,217,177)(38,142,224,170,44,136,218,176)(39,141,225,169,45,135,219,175)(40,140,226,180,46,134,220,174)(41,139,227,179,47,133,221,173)(42,138,228,178,48,144,222,172)(49,108,116,65,55,102,110,71)(50,107,117,64,56,101,111,70)(51,106,118,63,57,100,112,69)(52,105,119,62,58,99,113,68)(53,104,120,61,59,98,114,67)(54,103,109,72,60,97,115,66)(85,160,182,239,91,166,188,233)(86,159,183,238,92,165,189,232)(87,158,184,237,93,164,190,231)(88,157,185,236,94,163,191,230)(89,168,186,235,95,162,192,229)(90,167,187,234,96,161,181,240) );
G=PermutationGroup([[(1,80),(2,81),(3,82),(4,83),(5,84),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,121),(22,122),(23,123),(24,124),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,49),(35,50),(36,51),(37,161),(38,162),(39,163),(40,164),(41,165),(42,166),(43,167),(44,168),(45,157),(46,158),(47,159),(48,160),(61,211),(62,212),(63,213),(64,214),(65,215),(66,216),(67,205),(68,206),(69,207),(70,208),(71,209),(72,210),(85,178),(86,179),(87,180),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,201),(98,202),(99,203),(100,204),(101,193),(102,194),(103,195),(104,196),(105,197),(106,198),(107,199),(108,200),(109,153),(110,154),(111,155),(112,156),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(133,183),(134,184),(135,185),(136,186),(137,187),(138,188),(139,189),(140,190),(141,191),(142,192),(143,181),(144,182),(217,234),(218,235),(219,236),(220,237),(221,238),(222,239),(223,240),(224,229),(225,230),(226,231),(227,232),(228,233)], [(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)], [(1,71,174,181,203),(2,72,175,182,204),(3,61,176,183,193),(4,62,177,184,194),(5,63,178,185,195),(6,64,179,186,196),(7,65,180,187,197),(8,66,169,188,198),(9,67,170,189,199),(10,68,171,190,200),(11,69,172,191,201),(12,70,173,192,202),(13,236,57,153,48),(14,237,58,154,37),(15,238,59,155,38),(16,239,60,156,39),(17,240,49,145,40),(18,229,50,146,41),(19,230,51,147,42),(20,231,52,148,43),(21,232,53,149,44),(22,233,54,150,45),(23,234,55,151,46),(24,235,56,152,47),(25,116,167,132,226),(26,117,168,121,227),(27,118,157,122,228),(28,119,158,123,217),(29,120,159,124,218),(30,109,160,125,219),(31,110,161,126,220),(32,111,162,127,221),(33,112,163,128,222),(34,113,164,129,223),(35,114,165,130,224),(36,115,166,131,225),(73,214,86,136,104),(74,215,87,137,105),(75,216,88,138,106),(76,205,89,139,107),(77,206,90,140,108),(78,207,91,141,97),(79,208,92,142,98),(80,209,93,143,99),(81,210,94,144,100),(82,211,95,133,101),(83,212,96,134,102),(84,213,85,135,103)], [(1,17,77,132,7,23,83,126),(2,16,78,131,8,22,84,125),(3,15,79,130,9,21,73,124),(4,14,80,129,10,20,74,123),(5,13,81,128,11,19,75,122),(6,24,82,127,12,18,76,121),(25,197,151,212,31,203,145,206),(26,196,152,211,32,202,146,205),(27,195,153,210,33,201,147,216),(28,194,154,209,34,200,148,215),(29,193,155,208,35,199,149,214),(30,204,156,207,36,198,150,213),(37,143,223,171,43,137,217,177),(38,142,224,170,44,136,218,176),(39,141,225,169,45,135,219,175),(40,140,226,180,46,134,220,174),(41,139,227,179,47,133,221,173),(42,138,228,178,48,144,222,172),(49,108,116,65,55,102,110,71),(50,107,117,64,56,101,111,70),(51,106,118,63,57,100,112,69),(52,105,119,62,58,99,113,68),(53,104,120,61,59,98,114,67),(54,103,109,72,60,97,115,66),(85,160,182,239,91,166,188,233),(86,159,183,238,92,165,189,232),(87,158,184,237,93,164,190,231),(88,157,185,236,94,163,191,230),(89,168,186,235,95,162,192,229),(90,167,187,234,96,161,181,240)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | D6 | Dic3 | Dic3 | M4(2) | C4.Dic3 | F5 | C2×F5 | C2×F5 | C3⋊F5 | C4.F5 | C2×C3⋊F5 | C2×C3⋊F5 | C12.F5 |
| kernel | C2×C12.F5 | C12.F5 | C2×C15⋊C8 | D5×C2×C12 | D5×C12 | C2×C60 | D5×C2×C6 | C2×C4×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C2×C12.F5 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 84 | 0 | 224 | 224 |
| 0 | 0 | 17 | 101 | 17 | 0 |
| 0 | 0 | 0 | 17 | 101 | 17 |
| 0 | 0 | 224 | 224 | 0 | 84 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 81 | 21 | 0 | 0 | 0 | 0 |
| 181 | 160 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 54 | 211 | 84 |
| 0 | 0 | 157 | 30 | 187 | 214 |
| 0 | 0 | 157 | 184 | 211 | 127 |
| 0 | 0 | 27 | 184 | 57 | 214 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,240,0,0,0,0,1,0,0,0,0,0,0,0,84,17,0,224,0,0,0,101,17,224,0,0,224,17,101,0,0,0,224,0,17,84],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[81,181,0,0,0,0,21,160,0,0,0,0,0,0,27,157,157,27,0,0,54,30,184,184,0,0,211,187,211,57,0,0,84,214,127,214] >;
C2×C12.F5 in GAP, Magma, Sage, TeX
C_2\times C_{12}.F_5 % in TeX
G:=Group("C2xC12.F5"); // GroupNames label
G:=SmallGroup(480,1061);
// by ID
G=gap.SmallGroup(480,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^5=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations