direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20⋊D4, C60⋊18D4, C20⋊3(C3×D4), (C2×D20)⋊9C6, (D4×C10)⋊4C6, (C6×D4)⋊13D5, (C6×D20)⋊25C2, (D4×C30)⋊11C2, C12⋊9(C5⋊D4), C15⋊9(C4⋊1D4), (C4×Dic5)⋊6C6, Dic5⋊1(C3×D4), C10.41(C6×D4), C6.195(D4×D5), C23.9(C6×D5), (C3×Dic5)⋊10D4, C30.409(C2×D4), (C12×Dic5)⋊18C2, (C2×C12).364D10, (C22×C6).11D10, (C2×C30).372C23, (C2×C60).295C22, (C22×C30).108C22, (C6×Dic5).252C22, C5⋊2(C3×C4⋊1D4), C4⋊1(C3×C5⋊D4), C2.28(C3×D4×D5), (C2×D4)⋊6(C3×D5), (C2×C5⋊D4)⋊7C6, (C6×C5⋊D4)⋊22C2, (C2×C4).51(C6×D5), C2.16(C6×C5⋊D4), C22.62(D5×C2×C6), (C2×C20).32(C2×C6), C6.137(C2×C5⋊D4), (D5×C2×C6).84C22, (C22×C10).27(C2×C6), (C2×C10).55(C22×C6), (C2×Dic5).42(C2×C6), (C22×D5).14(C2×C6), (C2×C6).368(C22×D5), SmallGroup(480,733)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20⋊D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=c-1 >
Subgroups: 800 in 216 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C4⋊1D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4×C12, C6×D4, C6×D4, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4, D4×C10, C3×C4⋊1D4, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, C22×C30, C20⋊D4, C12×Dic5, C6×D20, C6×C5⋊D4, D4×C30, C3×C20⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C4⋊1D4, C5⋊D4, C22×D5, C6×D4, C6×D5, D4×D5, C2×C5⋊D4, C3×C4⋊1D4, C3×C5⋊D4, D5×C2×C6, C20⋊D4, C3×D4×D5, C6×C5⋊D4, C3×C20⋊D4
►On 240 points
Generators in S240
(1 112 188)(2 113 189)(3 114 190)(4 115 191)(5 116 192)(6 117 193)(7 118 194)(8 119 195)(9 120 196)(10 101 197)(11 102 198)(12 103 199)(13 104 200)(14 105 181)(15 106 182)(16 107 183)(17 108 184)(18 109 185)(19 110 186)(20 111 187)(21 145 131)(22 146 132)(23 147 133)(24 148 134)(25 149 135)(26 150 136)(27 151 137)(28 152 138)(29 153 139)(30 154 140)(31 155 121)(32 156 122)(33 157 123)(34 158 124)(35 159 125)(36 160 126)(37 141 127)(38 142 128)(39 143 129)(40 144 130)(41 172 98)(42 173 99)(43 174 100)(44 175 81)(45 176 82)(46 177 83)(47 178 84)(48 179 85)(49 180 86)(50 161 87)(51 162 88)(52 163 89)(53 164 90)(54 165 91)(55 166 92)(56 167 93)(57 168 94)(58 169 95)(59 170 96)(60 171 97)(61 233 219)(62 234 220)(63 235 201)(64 236 202)(65 237 203)(66 238 204)(67 239 205)(68 240 206)(69 221 207)(70 222 208)(71 223 209)(72 224 210)(73 225 211)(74 226 212)(75 227 213)(76 228 214)(77 229 215)(78 230 216)(79 231 217)(80 232 218)
(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)
(1 201 142 97)(2 210 143 86)(3 219 144 95)(4 208 145 84)(5 217 146 93)(6 206 147 82)(7 215 148 91)(8 204 149 100)(9 213 150 89)(10 202 151 98)(11 211 152 87)(12 220 153 96)(13 209 154 85)(14 218 155 94)(15 207 156 83)(16 216 157 92)(17 205 158 81)(18 214 159 90)(19 203 160 99)(20 212 141 88)(21 178 191 222)(22 167 192 231)(23 176 193 240)(24 165 194 229)(25 174 195 238)(26 163 196 227)(27 172 197 236)(28 161 198 225)(29 170 199 234)(30 179 200 223)(31 168 181 232)(32 177 182 221)(33 166 183 230)(34 175 184 239)(35 164 185 228)(36 173 186 237)(37 162 187 226)(38 171 188 235)(39 180 189 224)(40 169 190 233)(41 101 64 137)(42 110 65 126)(43 119 66 135)(44 108 67 124)(45 117 68 133)(46 106 69 122)(47 115 70 131)(48 104 71 140)(49 113 72 129)(50 102 73 138)(51 111 74 127)(52 120 75 136)(53 109 76 125)(54 118 77 134)(55 107 78 123)(56 116 79 132)(57 105 80 121)(58 114 61 130)(59 103 62 139)(60 112 63 128)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 80)(59 79)(60 78)(81 212)(82 211)(83 210)(84 209)(85 208)(86 207)(87 206)(88 205)(89 204)(90 203)(91 202)(92 201)(93 220)(94 219)(95 218)(96 217)(97 216)(98 215)(99 214)(100 213)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 240)(162 239)(163 238)(164 237)(165 236)(166 235)(167 234)(168 233)(169 232)(170 231)(171 230)(172 229)(173 228)(174 227)(175 226)(176 225)(177 224)(178 223)(179 222)(180 221)(181 190)(182 189)(183 188)(184 187)(185 186)(191 200)(192 199)(193 198)(194 197)(195 196)
G:=sub<Sym(240)| (1,112,188)(2,113,189)(3,114,190)(4,115,191)(5,116,192)(6,117,193)(7,118,194)(8,119,195)(9,120,196)(10,101,197)(11,102,198)(12,103,199)(13,104,200)(14,105,181)(15,106,182)(16,107,183)(17,108,184)(18,109,185)(19,110,186)(20,111,187)(21,145,131)(22,146,132)(23,147,133)(24,148,134)(25,149,135)(26,150,136)(27,151,137)(28,152,138)(29,153,139)(30,154,140)(31,155,121)(32,156,122)(33,157,123)(34,158,124)(35,159,125)(36,160,126)(37,141,127)(38,142,128)(39,143,129)(40,144,130)(41,172,98)(42,173,99)(43,174,100)(44,175,81)(45,176,82)(46,177,83)(47,178,84)(48,179,85)(49,180,86)(50,161,87)(51,162,88)(52,163,89)(53,164,90)(54,165,91)(55,166,92)(56,167,93)(57,168,94)(58,169,95)(59,170,96)(60,171,97)(61,233,219)(62,234,220)(63,235,201)(64,236,202)(65,237,203)(66,238,204)(67,239,205)(68,240,206)(69,221,207)(70,222,208)(71,223,209)(72,224,210)(73,225,211)(74,226,212)(75,227,213)(76,228,214)(77,229,215)(78,230,216)(79,231,217)(80,232,218), (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), (1,201,142,97)(2,210,143,86)(3,219,144,95)(4,208,145,84)(5,217,146,93)(6,206,147,82)(7,215,148,91)(8,204,149,100)(9,213,150,89)(10,202,151,98)(11,211,152,87)(12,220,153,96)(13,209,154,85)(14,218,155,94)(15,207,156,83)(16,216,157,92)(17,205,158,81)(18,214,159,90)(19,203,160,99)(20,212,141,88)(21,178,191,222)(22,167,192,231)(23,176,193,240)(24,165,194,229)(25,174,195,238)(26,163,196,227)(27,172,197,236)(28,161,198,225)(29,170,199,234)(30,179,200,223)(31,168,181,232)(32,177,182,221)(33,166,183,230)(34,175,184,239)(35,164,185,228)(36,173,186,237)(37,162,187,226)(38,171,188,235)(39,180,189,224)(40,169,190,233)(41,101,64,137)(42,110,65,126)(43,119,66,135)(44,108,67,124)(45,117,68,133)(46,106,69,122)(47,115,70,131)(48,104,71,140)(49,113,72,129)(50,102,73,138)(51,111,74,127)(52,120,75,136)(53,109,76,125)(54,118,77,134)(55,107,78,123)(56,116,79,132)(57,105,80,121)(58,114,61,130)(59,103,62,139)(60,112,63,128), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,80)(59,79)(60,78)(81,212)(82,211)(83,210)(84,209)(85,208)(86,207)(87,206)(88,205)(89,204)(90,203)(91,202)(92,201)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,240)(162,239)(163,238)(164,237)(165,236)(166,235)(167,234)(168,233)(169,232)(170,231)(171,230)(172,229)(173,228)(174,227)(175,226)(176,225)(177,224)(178,223)(179,222)(180,221)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)>;
G:=Group( (1,112,188)(2,113,189)(3,114,190)(4,115,191)(5,116,192)(6,117,193)(7,118,194)(8,119,195)(9,120,196)(10,101,197)(11,102,198)(12,103,199)(13,104,200)(14,105,181)(15,106,182)(16,107,183)(17,108,184)(18,109,185)(19,110,186)(20,111,187)(21,145,131)(22,146,132)(23,147,133)(24,148,134)(25,149,135)(26,150,136)(27,151,137)(28,152,138)(29,153,139)(30,154,140)(31,155,121)(32,156,122)(33,157,123)(34,158,124)(35,159,125)(36,160,126)(37,141,127)(38,142,128)(39,143,129)(40,144,130)(41,172,98)(42,173,99)(43,174,100)(44,175,81)(45,176,82)(46,177,83)(47,178,84)(48,179,85)(49,180,86)(50,161,87)(51,162,88)(52,163,89)(53,164,90)(54,165,91)(55,166,92)(56,167,93)(57,168,94)(58,169,95)(59,170,96)(60,171,97)(61,233,219)(62,234,220)(63,235,201)(64,236,202)(65,237,203)(66,238,204)(67,239,205)(68,240,206)(69,221,207)(70,222,208)(71,223,209)(72,224,210)(73,225,211)(74,226,212)(75,227,213)(76,228,214)(77,229,215)(78,230,216)(79,231,217)(80,232,218), (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), (1,201,142,97)(2,210,143,86)(3,219,144,95)(4,208,145,84)(5,217,146,93)(6,206,147,82)(7,215,148,91)(8,204,149,100)(9,213,150,89)(10,202,151,98)(11,211,152,87)(12,220,153,96)(13,209,154,85)(14,218,155,94)(15,207,156,83)(16,216,157,92)(17,205,158,81)(18,214,159,90)(19,203,160,99)(20,212,141,88)(21,178,191,222)(22,167,192,231)(23,176,193,240)(24,165,194,229)(25,174,195,238)(26,163,196,227)(27,172,197,236)(28,161,198,225)(29,170,199,234)(30,179,200,223)(31,168,181,232)(32,177,182,221)(33,166,183,230)(34,175,184,239)(35,164,185,228)(36,173,186,237)(37,162,187,226)(38,171,188,235)(39,180,189,224)(40,169,190,233)(41,101,64,137)(42,110,65,126)(43,119,66,135)(44,108,67,124)(45,117,68,133)(46,106,69,122)(47,115,70,131)(48,104,71,140)(49,113,72,129)(50,102,73,138)(51,111,74,127)(52,120,75,136)(53,109,76,125)(54,118,77,134)(55,107,78,123)(56,116,79,132)(57,105,80,121)(58,114,61,130)(59,103,62,139)(60,112,63,128), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,80)(59,79)(60,78)(81,212)(82,211)(83,210)(84,209)(85,208)(86,207)(87,206)(88,205)(89,204)(90,203)(91,202)(92,201)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,240)(162,239)(163,238)(164,237)(165,236)(166,235)(167,234)(168,233)(169,232)(170,231)(171,230)(172,229)(173,228)(174,227)(175,226)(176,225)(177,224)(178,223)(179,222)(180,221)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196) );
G=PermutationGroup([[(1,112,188),(2,113,189),(3,114,190),(4,115,191),(5,116,192),(6,117,193),(7,118,194),(8,119,195),(9,120,196),(10,101,197),(11,102,198),(12,103,199),(13,104,200),(14,105,181),(15,106,182),(16,107,183),(17,108,184),(18,109,185),(19,110,186),(20,111,187),(21,145,131),(22,146,132),(23,147,133),(24,148,134),(25,149,135),(26,150,136),(27,151,137),(28,152,138),(29,153,139),(30,154,140),(31,155,121),(32,156,122),(33,157,123),(34,158,124),(35,159,125),(36,160,126),(37,141,127),(38,142,128),(39,143,129),(40,144,130),(41,172,98),(42,173,99),(43,174,100),(44,175,81),(45,176,82),(46,177,83),(47,178,84),(48,179,85),(49,180,86),(50,161,87),(51,162,88),(52,163,89),(53,164,90),(54,165,91),(55,166,92),(56,167,93),(57,168,94),(58,169,95),(59,170,96),(60,171,97),(61,233,219),(62,234,220),(63,235,201),(64,236,202),(65,237,203),(66,238,204),(67,239,205),(68,240,206),(69,221,207),(70,222,208),(71,223,209),(72,224,210),(73,225,211),(74,226,212),(75,227,213),(76,228,214),(77,229,215),(78,230,216),(79,231,217),(80,232,218)], [(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)], [(1,201,142,97),(2,210,143,86),(3,219,144,95),(4,208,145,84),(5,217,146,93),(6,206,147,82),(7,215,148,91),(8,204,149,100),(9,213,150,89),(10,202,151,98),(11,211,152,87),(12,220,153,96),(13,209,154,85),(14,218,155,94),(15,207,156,83),(16,216,157,92),(17,205,158,81),(18,214,159,90),(19,203,160,99),(20,212,141,88),(21,178,191,222),(22,167,192,231),(23,176,193,240),(24,165,194,229),(25,174,195,238),(26,163,196,227),(27,172,197,236),(28,161,198,225),(29,170,199,234),(30,179,200,223),(31,168,181,232),(32,177,182,221),(33,166,183,230),(34,175,184,239),(35,164,185,228),(36,173,186,237),(37,162,187,226),(38,171,188,235),(39,180,189,224),(40,169,190,233),(41,101,64,137),(42,110,65,126),(43,119,66,135),(44,108,67,124),(45,117,68,133),(46,106,69,122),(47,115,70,131),(48,104,71,140),(49,113,72,129),(50,102,73,138),(51,111,74,127),(52,120,75,136),(53,109,76,125),(54,118,77,134),(55,107,78,123),(56,116,79,132),(57,105,80,121),(58,114,61,130),(59,103,62,139),(60,112,63,128)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,20),(18,19),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,77),(42,76),(43,75),(44,74),(45,73),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(57,61),(58,80),(59,79),(60,78),(81,212),(82,211),(83,210),(84,209),(85,208),(86,207),(87,206),(88,205),(89,204),(90,203),(91,202),(92,201),(93,220),(94,219),(95,218),(96,217),(97,216),(98,215),(99,214),(100,213),(101,118),(102,117),(103,116),(104,115),(105,114),(106,113),(107,112),(108,111),(109,110),(119,120),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150),(159,160),(161,240),(162,239),(163,238),(164,237),(165,236),(166,235),(167,234),(168,233),(169,232),(170,231),(171,230),(172,229),(173,228),(174,227),(175,226),(176,225),(177,224),(178,223),(179,222),(180,221),(181,190),(182,189),(183,188),(184,187),(185,186),(191,200),(192,199),(193,198),(194,197),(195,196)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | D4×D5 | C3×D4×D5 |
| kernel | C3×C20⋊D4 | C12×Dic5 | C6×D20 | C6×C5⋊D4 | D4×C30 | C20⋊D4 | C4×Dic5 | C2×D20 | C2×C5⋊D4 | D4×C10 | C3×Dic5 | C60 | C6×D4 | C2×C12 | C22×C6 | Dic5 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 2 | 8 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 8 | 4 | 8 | 16 | 4 | 8 |
Matrix representation of C3×C20⋊D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 |
| 1 | 44 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 60 | 60 |
| 14 | 45 | 0 | 0 |
| 39 | 47 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 44 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,44,0,0,0,0,1,60,0,0,2,60],[14,39,0,0,45,47,0,0,0,0,60,0,0,0,0,60],[60,44,0,0,0,1,0,0,0,0,1,0,0,0,2,60] >;
C3×C20⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}\rtimes D_4 % in TeX
G:=Group("C3xC20:D4"); // GroupNames label
G:=SmallGroup(480,733);
// by ID
G=gap.SmallGroup(480,733);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,555,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations