direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D20.3C4, D20.3C12, C24.84D10, C60.276C23, Dic10.3C12, C120.102C22, (C8×D5)⋊6C6, (C2×C40)⋊10C6, C8⋊D5⋊7C6, (C2×C24)⋊15D5, C8.18(C6×D5), (C2×C120)⋊23C2, (D5×C24)⋊15C2, C15⋊22(C8○D4), C40.23(C2×C6), C4○D20.6C6, (C3×D20).6C4, C4.10(D5×C12), C12.67(C4×D5), C5⋊D4.3C12, C20.41(C2×C12), C60.202(C2×C4), D10.3(C2×C12), C4.Dic5⋊11C6, C22.2(D5×C12), (C2×C12).428D10, C20.36(C22×C6), (C3×Dic10).6C4, Dic5.5(C2×C12), C30.185(C22×C4), (C2×C60).518C22, C10.27(C22×C12), C12.243(C22×D5), (D5×C12).118C22, C5⋊3(C3×C8○D4), (C2×C8)⋊7(C3×D5), C4.37(D5×C2×C6), C2.15(D5×C2×C12), C6.110(C2×C4×D5), (C2×C4).79(C6×D5), (C2×C6).25(C4×D5), (C3×C5⋊D4).6C4, (C3×C8⋊D5)⋊15C2, C5⋊2C8.11(C2×C6), (C6×D5).37(C2×C4), (C4×D5).23(C2×C6), (C2×C30).150(C2×C4), (C2×C10).35(C2×C12), (C2×C20).101(C2×C6), (C3×C4○D20).12C2, (C3×C4.Dic5)⋊23C2, (C3×C5⋊2C8).51C22, (C3×Dic5).45(C2×C4), SmallGroup(480,694)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D20.3C4
G = < a,b,c,d | a3=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
Subgroups: 320 in 124 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C24, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C24, C2×C24, C3×M4(2), C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, C3×C8○D4, C3×C5⋊2C8, C120, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D20.3C4, D5×C24, C3×C8⋊D5, C3×C4.Dic5, C2×C120, C3×C4○D20, C3×D20.3C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, D10, C2×C12, C22×C6, C3×D5, C8○D4, C4×D5, C22×D5, C22×C12, C6×D5, C2×C4×D5, C3×C8○D4, D5×C12, D5×C2×C6, D20.3C4, D5×C2×C12, C3×D20.3C4
►On 240 points
Generators in S240
(1 163 234)(2 164 235)(3 165 236)(4 166 237)(5 167 238)(6 168 239)(7 169 240)(8 170 221)(9 171 222)(10 172 223)(11 173 224)(12 174 225)(13 175 226)(14 176 227)(15 177 228)(16 178 229)(17 179 230)(18 180 231)(19 161 232)(20 162 233)(21 76 197)(22 77 198)(23 78 199)(24 79 200)(25 80 181)(26 61 182)(27 62 183)(28 63 184)(29 64 185)(30 65 186)(31 66 187)(32 67 188)(33 68 189)(34 69 190)(35 70 191)(36 71 192)(37 72 193)(38 73 194)(39 74 195)(40 75 196)(41 83 151)(42 84 152)(43 85 153)(44 86 154)(45 87 155)(46 88 156)(47 89 157)(48 90 158)(49 91 159)(50 92 160)(51 93 141)(52 94 142)(53 95 143)(54 96 144)(55 97 145)(56 98 146)(57 99 147)(58 100 148)(59 81 149)(60 82 150)(101 130 214)(102 131 215)(103 132 216)(104 133 217)(105 134 218)(106 135 219)(107 136 220)(108 137 201)(109 138 202)(110 139 203)(111 140 204)(112 121 205)(113 122 206)(114 123 207)(115 124 208)(116 125 209)(117 126 210)(118 127 211)(119 128 212)(120 129 213)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 122)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)(161 164)(162 163)(165 180)(166 179)(167 178)(168 177)(169 176)(170 175)(171 174)(172 173)(181 188)(182 187)(183 186)(184 185)(189 200)(190 199)(191 198)(192 197)(193 196)(194 195)(201 210)(202 209)(203 208)(204 207)(205 206)(211 220)(212 219)(213 218)(214 217)(215 216)(221 226)(222 225)(223 224)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)
(1 159 122 39 11 149 132 29)(2 160 123 40 12 150 133 30)(3 141 124 21 13 151 134 31)(4 142 125 22 14 152 135 32)(5 143 126 23 15 153 136 33)(6 144 127 24 16 154 137 34)(7 145 128 25 17 155 138 35)(8 146 129 26 18 156 139 36)(9 147 130 27 19 157 140 37)(10 148 131 28 20 158 121 38)(41 218 66 165 51 208 76 175)(42 219 67 166 52 209 77 176)(43 220 68 167 53 210 78 177)(44 201 69 168 54 211 79 178)(45 202 70 169 55 212 80 179)(46 203 71 170 56 213 61 180)(47 204 72 171 57 214 62 161)(48 205 73 172 58 215 63 162)(49 206 74 173 59 216 64 163)(50 207 75 174 60 217 65 164)(81 103 185 234 91 113 195 224)(82 104 186 235 92 114 196 225)(83 105 187 236 93 115 197 226)(84 106 188 237 94 116 198 227)(85 107 189 238 95 117 199 228)(86 108 190 239 96 118 200 229)(87 109 191 240 97 119 181 230)(88 110 192 221 98 120 182 231)(89 111 193 222 99 101 183 232)(90 112 194 223 100 102 184 233)
G:=sub<Sym(240)| (1,163,234)(2,164,235)(3,165,236)(4,166,237)(5,167,238)(6,168,239)(7,169,240)(8,170,221)(9,171,222)(10,172,223)(11,173,224)(12,174,225)(13,175,226)(14,176,227)(15,177,228)(16,178,229)(17,179,230)(18,180,231)(19,161,232)(20,162,233)(21,76,197)(22,77,198)(23,78,199)(24,79,200)(25,80,181)(26,61,182)(27,62,183)(28,63,184)(29,64,185)(30,65,186)(31,66,187)(32,67,188)(33,68,189)(34,69,190)(35,70,191)(36,71,192)(37,72,193)(38,73,194)(39,74,195)(40,75,196)(41,83,151)(42,84,152)(43,85,153)(44,86,154)(45,87,155)(46,88,156)(47,89,157)(48,90,158)(49,91,159)(50,92,160)(51,93,141)(52,94,142)(53,95,143)(54,96,144)(55,97,145)(56,98,146)(57,99,147)(58,100,148)(59,81,149)(60,82,150)(101,130,214)(102,131,215)(103,132,216)(104,133,217)(105,134,218)(106,135,219)(107,136,220)(108,137,201)(109,138,202)(110,139,203)(111,140,204)(112,121,205)(113,122,206)(114,123,207)(115,124,208)(116,125,209)(117,126,210)(118,127,211)(119,128,212)(120,129,213), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,122)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,164)(162,163)(165,180)(166,179)(167,178)(168,177)(169,176)(170,175)(171,174)(172,173)(181,188)(182,187)(183,186)(184,185)(189,200)(190,199)(191,198)(192,197)(193,196)(194,195)(201,210)(202,209)(203,208)(204,207)(205,206)(211,220)(212,219)(213,218)(214,217)(215,216)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,159,122,39,11,149,132,29)(2,160,123,40,12,150,133,30)(3,141,124,21,13,151,134,31)(4,142,125,22,14,152,135,32)(5,143,126,23,15,153,136,33)(6,144,127,24,16,154,137,34)(7,145,128,25,17,155,138,35)(8,146,129,26,18,156,139,36)(9,147,130,27,19,157,140,37)(10,148,131,28,20,158,121,38)(41,218,66,165,51,208,76,175)(42,219,67,166,52,209,77,176)(43,220,68,167,53,210,78,177)(44,201,69,168,54,211,79,178)(45,202,70,169,55,212,80,179)(46,203,71,170,56,213,61,180)(47,204,72,171,57,214,62,161)(48,205,73,172,58,215,63,162)(49,206,74,173,59,216,64,163)(50,207,75,174,60,217,65,164)(81,103,185,234,91,113,195,224)(82,104,186,235,92,114,196,225)(83,105,187,236,93,115,197,226)(84,106,188,237,94,116,198,227)(85,107,189,238,95,117,199,228)(86,108,190,239,96,118,200,229)(87,109,191,240,97,119,181,230)(88,110,192,221,98,120,182,231)(89,111,193,222,99,101,183,232)(90,112,194,223,100,102,184,233)>;
G:=Group( (1,163,234)(2,164,235)(3,165,236)(4,166,237)(5,167,238)(6,168,239)(7,169,240)(8,170,221)(9,171,222)(10,172,223)(11,173,224)(12,174,225)(13,175,226)(14,176,227)(15,177,228)(16,178,229)(17,179,230)(18,180,231)(19,161,232)(20,162,233)(21,76,197)(22,77,198)(23,78,199)(24,79,200)(25,80,181)(26,61,182)(27,62,183)(28,63,184)(29,64,185)(30,65,186)(31,66,187)(32,67,188)(33,68,189)(34,69,190)(35,70,191)(36,71,192)(37,72,193)(38,73,194)(39,74,195)(40,75,196)(41,83,151)(42,84,152)(43,85,153)(44,86,154)(45,87,155)(46,88,156)(47,89,157)(48,90,158)(49,91,159)(50,92,160)(51,93,141)(52,94,142)(53,95,143)(54,96,144)(55,97,145)(56,98,146)(57,99,147)(58,100,148)(59,81,149)(60,82,150)(101,130,214)(102,131,215)(103,132,216)(104,133,217)(105,134,218)(106,135,219)(107,136,220)(108,137,201)(109,138,202)(110,139,203)(111,140,204)(112,121,205)(113,122,206)(114,123,207)(115,124,208)(116,125,209)(117,126,210)(118,127,211)(119,128,212)(120,129,213), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,122)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,164)(162,163)(165,180)(166,179)(167,178)(168,177)(169,176)(170,175)(171,174)(172,173)(181,188)(182,187)(183,186)(184,185)(189,200)(190,199)(191,198)(192,197)(193,196)(194,195)(201,210)(202,209)(203,208)(204,207)(205,206)(211,220)(212,219)(213,218)(214,217)(215,216)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,159,122,39,11,149,132,29)(2,160,123,40,12,150,133,30)(3,141,124,21,13,151,134,31)(4,142,125,22,14,152,135,32)(5,143,126,23,15,153,136,33)(6,144,127,24,16,154,137,34)(7,145,128,25,17,155,138,35)(8,146,129,26,18,156,139,36)(9,147,130,27,19,157,140,37)(10,148,131,28,20,158,121,38)(41,218,66,165,51,208,76,175)(42,219,67,166,52,209,77,176)(43,220,68,167,53,210,78,177)(44,201,69,168,54,211,79,178)(45,202,70,169,55,212,80,179)(46,203,71,170,56,213,61,180)(47,204,72,171,57,214,62,161)(48,205,73,172,58,215,63,162)(49,206,74,173,59,216,64,163)(50,207,75,174,60,217,65,164)(81,103,185,234,91,113,195,224)(82,104,186,235,92,114,196,225)(83,105,187,236,93,115,197,226)(84,106,188,237,94,116,198,227)(85,107,189,238,95,117,199,228)(86,108,190,239,96,118,200,229)(87,109,191,240,97,119,181,230)(88,110,192,221,98,120,182,231)(89,111,193,222,99,101,183,232)(90,112,194,223,100,102,184,233) );
G=PermutationGroup([[(1,163,234),(2,164,235),(3,165,236),(4,166,237),(5,167,238),(6,168,239),(7,169,240),(8,170,221),(9,171,222),(10,172,223),(11,173,224),(12,174,225),(13,175,226),(14,176,227),(15,177,228),(16,178,229),(17,179,230),(18,180,231),(19,161,232),(20,162,233),(21,76,197),(22,77,198),(23,78,199),(24,79,200),(25,80,181),(26,61,182),(27,62,183),(28,63,184),(29,64,185),(30,65,186),(31,66,187),(32,67,188),(33,68,189),(34,69,190),(35,70,191),(36,71,192),(37,72,193),(38,73,194),(39,74,195),(40,75,196),(41,83,151),(42,84,152),(43,85,153),(44,86,154),(45,87,155),(46,88,156),(47,89,157),(48,90,158),(49,91,159),(50,92,160),(51,93,141),(52,94,142),(53,95,143),(54,96,144),(55,97,145),(56,98,146),(57,99,147),(58,100,148),(59,81,149),(60,82,150),(101,130,214),(102,131,215),(103,132,216),(104,133,217),(105,134,218),(106,135,219),(107,136,220),(108,137,201),(109,138,202),(110,139,203),(111,140,204),(112,121,205),(113,122,206),(114,123,207),(115,124,208),(116,125,209),(117,126,210),(118,127,211),(119,128,212),(120,129,213)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(121,122),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,160),(158,159),(161,164),(162,163),(165,180),(166,179),(167,178),(168,177),(169,176),(170,175),(171,174),(172,173),(181,188),(182,187),(183,186),(184,185),(189,200),(190,199),(191,198),(192,197),(193,196),(194,195),(201,210),(202,209),(203,208),(204,207),(205,206),(211,220),(212,219),(213,218),(214,217),(215,216),(221,226),(222,225),(223,224),(227,240),(228,239),(229,238),(230,237),(231,236),(232,235),(233,234)], [(1,159,122,39,11,149,132,29),(2,160,123,40,12,150,133,30),(3,141,124,21,13,151,134,31),(4,142,125,22,14,152,135,32),(5,143,126,23,15,153,136,33),(6,144,127,24,16,154,137,34),(7,145,128,25,17,155,138,35),(8,146,129,26,18,156,139,36),(9,147,130,27,19,157,140,37),(10,148,131,28,20,158,121,38),(41,218,66,165,51,208,76,175),(42,219,67,166,52,209,77,176),(43,220,68,167,53,210,78,177),(44,201,69,168,54,211,79,178),(45,202,70,169,55,212,80,179),(46,203,71,170,56,213,61,180),(47,204,72,171,57,214,62,161),(48,205,73,172,58,215,63,162),(49,206,74,173,59,216,64,163),(50,207,75,174,60,217,65,164),(81,103,185,234,91,113,195,224),(82,104,186,235,92,114,196,225),(83,105,187,236,93,115,197,226),(84,106,188,237,94,116,198,227),(85,107,189,238,95,117,199,228),(86,108,190,239,96,118,200,229),(87,109,191,240,97,119,181,230),(88,110,192,221,98,120,182,231),(89,111,193,222,99,101,183,232),(90,112,194,223,100,102,184,233)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 24M | ··· | 24T | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D5 | D10 | D10 | C3×D5 | C8○D4 | C4×D5 | C4×D5 | C6×D5 | C6×D5 | C3×C8○D4 | D5×C12 | D5×C12 | D20.3C4 | C3×D20.3C4 |
| kernel | C3×D20.3C4 | D5×C24 | C3×C8⋊D5 | C3×C4.Dic5 | C2×C120 | C3×C4○D20 | D20.3C4 | C3×Dic10 | C3×D20 | C3×C5⋊D4 | C8×D5 | C8⋊D5 | C4.Dic5 | C2×C40 | C4○D20 | Dic10 | D20 | C5⋊D4 | C2×C24 | C24 | C2×C12 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C3×D20.3C4 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 238 | 44 |
| 197 | 163 |
| 163 | 163 |
| 44 | 78 |
| 211 | 0 |
| 0 | 211 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[238,197,44,163],[163,44,163,78],[211,0,0,211] >;
C3×D20.3C4 in GAP, Magma, Sage, TeX
C_3\times D_{20}._3C_4 % in TeX
G:=Group("C3xD20.3C4"); // GroupNames label
G:=SmallGroup(480,694);
// by ID
G=gap.SmallGroup(480,694);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,142,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations