direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C3⋊D20, C60⋊14D4, C12⋊8D20, C3⋊4(C4×D20), C15⋊19(C4×D4), D10⋊9(C4×S3), D30⋊24(C2×C4), C20⋊8(C3⋊D4), Dic3⋊4(C4×D5), C6.60(C2×D20), (C4×Dic3)⋊16D5, C30.143(C2×D4), (C2×C20).340D6, D30⋊4C4⋊39C2, (Dic3×C20)⋊12C2, C6.35(C4○D20), C30.78(C4○D4), (C2×C12).344D10, C30.62(C22×C4), C6.Dic10⋊43C2, (C22×D5).87D6, C10.38(C4○D12), D10⋊Dic3⋊40C2, (C2×C60).242C22, (C2×C30).133C23, (C2×Dic5).179D6, (C2×Dic3).152D10, C2.5(D6.D10), (C6×Dic5).205C22, (C2×Dic15).211C22, (C10×Dic3).185C22, (C22×D15).106C22, C5⋊2(C4×C3⋊D4), (C2×C4×D5)⋊10S3, (D5×C2×C12)⋊8C2, C2.32(C4×S3×D5), C6.30(C2×C4×D5), (C2×C4×D15)⋊28C2, C10.63(S3×C2×C4), (C6×D5)⋊19(C2×C4), C2.1(C2×C3⋊D20), C22.65(C2×S3×D5), (C2×C4).245(S3×D5), C10.15(C2×C3⋊D4), (C5×Dic3)⋊16(C2×C4), (C2×C3⋊D20).11C2, (D5×C2×C6).103C22, (C2×C6).145(C22×D5), (C2×C10).145(C22×S3), SmallGroup(480,519)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3⋊D20
G = < a,b,c,d | a4=b3=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 988 in 188 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×C4×D5, C2×D20, C4×C3⋊D4, C3⋊D20, D5×C12, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4×D20, D10⋊Dic3, D30⋊4C4, C6.Dic10, Dic3×C20, C2×C3⋊D20, D5×C2×C12, C2×C4×D15, C4×C3⋊D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C3⋊D4, C22×S3, C4×D4, C4×D5, D20, C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, C2×D20, C4○D20, C4×C3⋊D4, C3⋊D20, C2×S3×D5, C4×D20, D6.D10, C4×S3×D5, C2×C3⋊D20, C4×C3⋊D20
►On 240 points
Generators in S240
(1 197 48 104)(2 198 49 105)(3 199 50 106)(4 200 51 107)(5 181 52 108)(6 182 53 109)(7 183 54 110)(8 184 55 111)(9 185 56 112)(10 186 57 113)(11 187 58 114)(12 188 59 115)(13 189 60 116)(14 190 41 117)(15 191 42 118)(16 192 43 119)(17 193 44 120)(18 194 45 101)(19 195 46 102)(20 196 47 103)(21 78 131 82)(22 79 132 83)(23 80 133 84)(24 61 134 85)(25 62 135 86)(26 63 136 87)(27 64 137 88)(28 65 138 89)(29 66 139 90)(30 67 140 91)(31 68 121 92)(32 69 122 93)(33 70 123 94)(34 71 124 95)(35 72 125 96)(36 73 126 97)(37 74 127 98)(38 75 128 99)(39 76 129 100)(40 77 130 81)(141 175 209 238)(142 176 210 239)(143 177 211 240)(144 178 212 221)(145 179 213 222)(146 180 214 223)(147 161 215 224)(148 162 216 225)(149 163 217 226)(150 164 218 227)(151 165 219 228)(152 166 220 229)(153 167 201 230)(154 168 202 231)(155 169 203 232)(156 170 204 233)(157 171 205 234)(158 172 206 235)(159 173 207 236)(160 174 208 237)
(1 128 208)(2 209 129)(3 130 210)(4 211 131)(5 132 212)(6 213 133)(7 134 214)(8 215 135)(9 136 216)(10 217 137)(11 138 218)(12 219 139)(13 140 220)(14 201 121)(15 122 202)(16 203 123)(17 124 204)(18 205 125)(19 126 206)(20 207 127)(21 51 143)(22 144 52)(23 53 145)(24 146 54)(25 55 147)(26 148 56)(27 57 149)(28 150 58)(29 59 151)(30 152 60)(31 41 153)(32 154 42)(33 43 155)(34 156 44)(35 45 157)(36 158 46)(37 47 159)(38 160 48)(39 49 141)(40 142 50)(61 180 110)(62 111 161)(63 162 112)(64 113 163)(65 164 114)(66 115 165)(67 166 116)(68 117 167)(69 168 118)(70 119 169)(71 170 120)(72 101 171)(73 172 102)(74 103 173)(75 174 104)(76 105 175)(77 176 106)(78 107 177)(79 178 108)(80 109 179)(81 239 199)(82 200 240)(83 221 181)(84 182 222)(85 223 183)(86 184 224)(87 225 185)(88 186 226)(89 227 187)(90 188 228)(91 229 189)(92 190 230)(93 231 191)(94 192 232)(95 233 193)(96 194 234)(97 235 195)(98 196 236)(99 237 197)(100 198 238)
(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 42)(2 41)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 219)(22 218)(23 217)(24 216)(25 215)(26 214)(27 213)(28 212)(29 211)(30 210)(31 209)(32 208)(33 207)(34 206)(35 205)(36 204)(37 203)(38 202)(39 201)(40 220)(61 225)(62 224)(63 223)(64 222)(65 221)(66 240)(67 239)(68 238)(69 237)(70 236)(71 235)(72 234)(73 233)(74 232)(75 231)(76 230)(77 229)(78 228)(79 227)(80 226)(81 166)(82 165)(83 164)(84 163)(85 162)(86 161)(87 180)(88 179)(89 178)(90 177)(91 176)(92 175)(93 174)(94 173)(95 172)(96 171)(97 170)(98 169)(99 168)(100 167)(101 194)(102 193)(103 192)(104 191)(105 190)(106 189)(107 188)(108 187)(109 186)(110 185)(111 184)(112 183)(113 182)(114 181)(115 200)(116 199)(117 198)(118 197)(119 196)(120 195)(121 141)(122 160)(123 159)(124 158)(125 157)(126 156)(127 155)(128 154)(129 153)(130 152)(131 151)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)
G:=sub<Sym(240)| (1,197,48,104)(2,198,49,105)(3,199,50,106)(4,200,51,107)(5,181,52,108)(6,182,53,109)(7,183,54,110)(8,184,55,111)(9,185,56,112)(10,186,57,113)(11,187,58,114)(12,188,59,115)(13,189,60,116)(14,190,41,117)(15,191,42,118)(16,192,43,119)(17,193,44,120)(18,194,45,101)(19,195,46,102)(20,196,47,103)(21,78,131,82)(22,79,132,83)(23,80,133,84)(24,61,134,85)(25,62,135,86)(26,63,136,87)(27,64,137,88)(28,65,138,89)(29,66,139,90)(30,67,140,91)(31,68,121,92)(32,69,122,93)(33,70,123,94)(34,71,124,95)(35,72,125,96)(36,73,126,97)(37,74,127,98)(38,75,128,99)(39,76,129,100)(40,77,130,81)(141,175,209,238)(142,176,210,239)(143,177,211,240)(144,178,212,221)(145,179,213,222)(146,180,214,223)(147,161,215,224)(148,162,216,225)(149,163,217,226)(150,164,218,227)(151,165,219,228)(152,166,220,229)(153,167,201,230)(154,168,202,231)(155,169,203,232)(156,170,204,233)(157,171,205,234)(158,172,206,235)(159,173,207,236)(160,174,208,237), (1,128,208)(2,209,129)(3,130,210)(4,211,131)(5,132,212)(6,213,133)(7,134,214)(8,215,135)(9,136,216)(10,217,137)(11,138,218)(12,219,139)(13,140,220)(14,201,121)(15,122,202)(16,203,123)(17,124,204)(18,205,125)(19,126,206)(20,207,127)(21,51,143)(22,144,52)(23,53,145)(24,146,54)(25,55,147)(26,148,56)(27,57,149)(28,150,58)(29,59,151)(30,152,60)(31,41,153)(32,154,42)(33,43,155)(34,156,44)(35,45,157)(36,158,46)(37,47,159)(38,160,48)(39,49,141)(40,142,50)(61,180,110)(62,111,161)(63,162,112)(64,113,163)(65,164,114)(66,115,165)(67,166,116)(68,117,167)(69,168,118)(70,119,169)(71,170,120)(72,101,171)(73,172,102)(74,103,173)(75,174,104)(76,105,175)(77,176,106)(78,107,177)(79,178,108)(80,109,179)(81,239,199)(82,200,240)(83,221,181)(84,182,222)(85,223,183)(86,184,224)(87,225,185)(88,186,226)(89,227,187)(90,188,228)(91,229,189)(92,190,230)(93,231,191)(94,192,232)(95,233,193)(96,194,234)(97,235,195)(98,196,236)(99,237,197)(100,198,238), (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), (1,42)(2,41)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,220)(61,225)(62,224)(63,223)(64,222)(65,221)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,180)(88,179)(89,178)(90,177)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,194)(102,193)(103,192)(104,191)(105,190)(106,189)(107,188)(108,187)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,141)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)>;
G:=Group( (1,197,48,104)(2,198,49,105)(3,199,50,106)(4,200,51,107)(5,181,52,108)(6,182,53,109)(7,183,54,110)(8,184,55,111)(9,185,56,112)(10,186,57,113)(11,187,58,114)(12,188,59,115)(13,189,60,116)(14,190,41,117)(15,191,42,118)(16,192,43,119)(17,193,44,120)(18,194,45,101)(19,195,46,102)(20,196,47,103)(21,78,131,82)(22,79,132,83)(23,80,133,84)(24,61,134,85)(25,62,135,86)(26,63,136,87)(27,64,137,88)(28,65,138,89)(29,66,139,90)(30,67,140,91)(31,68,121,92)(32,69,122,93)(33,70,123,94)(34,71,124,95)(35,72,125,96)(36,73,126,97)(37,74,127,98)(38,75,128,99)(39,76,129,100)(40,77,130,81)(141,175,209,238)(142,176,210,239)(143,177,211,240)(144,178,212,221)(145,179,213,222)(146,180,214,223)(147,161,215,224)(148,162,216,225)(149,163,217,226)(150,164,218,227)(151,165,219,228)(152,166,220,229)(153,167,201,230)(154,168,202,231)(155,169,203,232)(156,170,204,233)(157,171,205,234)(158,172,206,235)(159,173,207,236)(160,174,208,237), (1,128,208)(2,209,129)(3,130,210)(4,211,131)(5,132,212)(6,213,133)(7,134,214)(8,215,135)(9,136,216)(10,217,137)(11,138,218)(12,219,139)(13,140,220)(14,201,121)(15,122,202)(16,203,123)(17,124,204)(18,205,125)(19,126,206)(20,207,127)(21,51,143)(22,144,52)(23,53,145)(24,146,54)(25,55,147)(26,148,56)(27,57,149)(28,150,58)(29,59,151)(30,152,60)(31,41,153)(32,154,42)(33,43,155)(34,156,44)(35,45,157)(36,158,46)(37,47,159)(38,160,48)(39,49,141)(40,142,50)(61,180,110)(62,111,161)(63,162,112)(64,113,163)(65,164,114)(66,115,165)(67,166,116)(68,117,167)(69,168,118)(70,119,169)(71,170,120)(72,101,171)(73,172,102)(74,103,173)(75,174,104)(76,105,175)(77,176,106)(78,107,177)(79,178,108)(80,109,179)(81,239,199)(82,200,240)(83,221,181)(84,182,222)(85,223,183)(86,184,224)(87,225,185)(88,186,226)(89,227,187)(90,188,228)(91,229,189)(92,190,230)(93,231,191)(94,192,232)(95,233,193)(96,194,234)(97,235,195)(98,196,236)(99,237,197)(100,198,238), (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), (1,42)(2,41)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,219)(22,218)(23,217)(24,216)(25,215)(26,214)(27,213)(28,212)(29,211)(30,210)(31,209)(32,208)(33,207)(34,206)(35,205)(36,204)(37,203)(38,202)(39,201)(40,220)(61,225)(62,224)(63,223)(64,222)(65,221)(66,240)(67,239)(68,238)(69,237)(70,236)(71,235)(72,234)(73,233)(74,232)(75,231)(76,230)(77,229)(78,228)(79,227)(80,226)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,180)(88,179)(89,178)(90,177)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,194)(102,193)(103,192)(104,191)(105,190)(106,189)(107,188)(108,187)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,141)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142) );
G=PermutationGroup([[(1,197,48,104),(2,198,49,105),(3,199,50,106),(4,200,51,107),(5,181,52,108),(6,182,53,109),(7,183,54,110),(8,184,55,111),(9,185,56,112),(10,186,57,113),(11,187,58,114),(12,188,59,115),(13,189,60,116),(14,190,41,117),(15,191,42,118),(16,192,43,119),(17,193,44,120),(18,194,45,101),(19,195,46,102),(20,196,47,103),(21,78,131,82),(22,79,132,83),(23,80,133,84),(24,61,134,85),(25,62,135,86),(26,63,136,87),(27,64,137,88),(28,65,138,89),(29,66,139,90),(30,67,140,91),(31,68,121,92),(32,69,122,93),(33,70,123,94),(34,71,124,95),(35,72,125,96),(36,73,126,97),(37,74,127,98),(38,75,128,99),(39,76,129,100),(40,77,130,81),(141,175,209,238),(142,176,210,239),(143,177,211,240),(144,178,212,221),(145,179,213,222),(146,180,214,223),(147,161,215,224),(148,162,216,225),(149,163,217,226),(150,164,218,227),(151,165,219,228),(152,166,220,229),(153,167,201,230),(154,168,202,231),(155,169,203,232),(156,170,204,233),(157,171,205,234),(158,172,206,235),(159,173,207,236),(160,174,208,237)], [(1,128,208),(2,209,129),(3,130,210),(4,211,131),(5,132,212),(6,213,133),(7,134,214),(8,215,135),(9,136,216),(10,217,137),(11,138,218),(12,219,139),(13,140,220),(14,201,121),(15,122,202),(16,203,123),(17,124,204),(18,205,125),(19,126,206),(20,207,127),(21,51,143),(22,144,52),(23,53,145),(24,146,54),(25,55,147),(26,148,56),(27,57,149),(28,150,58),(29,59,151),(30,152,60),(31,41,153),(32,154,42),(33,43,155),(34,156,44),(35,45,157),(36,158,46),(37,47,159),(38,160,48),(39,49,141),(40,142,50),(61,180,110),(62,111,161),(63,162,112),(64,113,163),(65,164,114),(66,115,165),(67,166,116),(68,117,167),(69,168,118),(70,119,169),(71,170,120),(72,101,171),(73,172,102),(74,103,173),(75,174,104),(76,105,175),(77,176,106),(78,107,177),(79,178,108),(80,109,179),(81,239,199),(82,200,240),(83,221,181),(84,182,222),(85,223,183),(86,184,224),(87,225,185),(88,186,226),(89,227,187),(90,188,228),(91,229,189),(92,190,230),(93,231,191),(94,192,232),(95,233,193),(96,194,234),(97,235,195),(98,196,236),(99,237,197),(100,198,238)], [(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)], [(1,42),(2,41),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,219),(22,218),(23,217),(24,216),(25,215),(26,214),(27,213),(28,212),(29,211),(30,210),(31,209),(32,208),(33,207),(34,206),(35,205),(36,204),(37,203),(38,202),(39,201),(40,220),(61,225),(62,224),(63,223),(64,222),(65,221),(66,240),(67,239),(68,238),(69,237),(70,236),(71,235),(72,234),(73,233),(74,232),(75,231),(76,230),(77,229),(78,228),(79,227),(80,226),(81,166),(82,165),(83,164),(84,163),(85,162),(86,161),(87,180),(88,179),(89,178),(90,177),(91,176),(92,175),(93,174),(94,173),(95,172),(96,171),(97,170),(98,169),(99,168),(100,167),(101,194),(102,193),(103,192),(104,191),(105,190),(106,189),(107,188),(108,187),(109,186),(110,185),(111,184),(112,183),(113,182),(114,181),(115,200),(116,199),(117,198),(118,197),(119,196),(120,195),(121,141),(122,160),(123,159),(124,158),(125,157),(126,156),(127,155),(128,154),(129,153),(130,152),(131,151),(132,150),(133,149),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C4×S3 | C4×D5 | D20 | C4○D12 | C4○D20 | S3×D5 | C3⋊D20 | C2×S3×D5 | D6.D10 | C4×S3×D5 |
| kernel | C4×C3⋊D20 | D10⋊Dic3 | D30⋊4C4 | C6.Dic10 | Dic3×C20 | C2×C3⋊D20 | D5×C2×C12 | C2×C4×D15 | C3⋊D20 | C2×C4×D5 | C60 | C4×Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C20 | D10 | Dic3 | C12 | C10 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C4×C3⋊D20 ►in GL4(𝔽61) generated by
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 60 |
| 59 | 32 | 0 | 0 |
| 29 | 54 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 17 | 60 | 0 | 0 |
| 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[59,29,0,0,32,54,0,0,0,0,0,60,0,0,60,0],[17,44,0,0,60,44,0,0,0,0,0,60,0,0,60,0] >;
C4×C3⋊D20 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes D_{20} % in TeX
G:=Group("C4xC3:D20"); // GroupNames label
G:=SmallGroup(480,519);
// by ID
G=gap.SmallGroup(480,519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations