direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D20.2C4, D20.2C12, C24.64D10, C60.278C23, C120.75C22, Dic10.2C12, (C8×D5)⋊8C6, C8⋊D5⋊6C6, C4.5(D5×C12), C8.12(C6×D5), (D5×C24)⋊17C2, C15⋊25(C8○D4), C40.12(C2×C6), C4○D20.3C6, (C3×D20).5C4, C12.48(C4×D5), C5⋊D4.2C12, C60.168(C2×C4), C20.34(C2×C12), D10.4(C2×C12), M4(2)⋊5(C3×D5), (C5×M4(2))⋊4C6, C22.1(D5×C12), (C2×C12).358D10, (C3×M4(2))⋊11D5, C20.38(C22×C6), Dic5.6(C2×C12), (C3×Dic10).5C4, (C15×M4(2))⋊10C2, C10.29(C22×C12), C30.187(C22×C4), (C2×C60).286C22, C12.245(C22×D5), (D5×C12).120C22, C5⋊4(C3×C8○D4), (C2×C5⋊2C8)⋊3C6, C4.39(D5×C2×C6), C2.17(D5×C2×C12), (C6×C5⋊2C8)⋊17C2, C6.112(C2×C4×D5), (C2×C4).45(C6×D5), (C2×C6).15(C4×D5), (C3×C5⋊D4).5C4, (C3×C8⋊D5)⋊14C2, (C2×C20).23(C2×C6), C5⋊2C8.12(C2×C6), (C3×C4○D20).9C2, (C4×D5).24(C2×C6), (C6×D5).38(C2×C4), (C2×C30).123(C2×C4), (C2×C10).26(C2×C12), (C3×C5⋊2C8).61C22, (C3×Dic5).46(C2×C4), SmallGroup(480,700)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D20.2C4
G = < a,b,c,d | a3=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >
Subgroups: 320 in 124 conjugacy classes, 74 normal (42 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, M4(2), 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, C3×M4(2), C3×M4(2), C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, C3×C8○D4, C3×C5⋊2C8, C120, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D20.2C4, D5×C24, C3×C8⋊D5, C6×C5⋊2C8, C15×M4(2), C3×C4○D20, C3×D20.2C4
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, D20.2C4, D5×C2×C12, C3×D20.2C4
►On 240 points
Generators in S240
(1 240 155)(2 221 156)(3 222 157)(4 223 158)(5 224 159)(6 225 160)(7 226 141)(8 227 142)(9 228 143)(10 229 144)(11 230 145)(12 231 146)(13 232 147)(14 233 148)(15 234 149)(16 235 150)(17 236 151)(18 237 152)(19 238 153)(20 239 154)(21 165 81)(22 166 82)(23 167 83)(24 168 84)(25 169 85)(26 170 86)(27 171 87)(28 172 88)(29 173 89)(30 174 90)(31 175 91)(32 176 92)(33 177 93)(34 178 94)(35 179 95)(36 180 96)(37 161 97)(38 162 98)(39 163 99)(40 164 100)(41 199 204)(42 200 205)(43 181 206)(44 182 207)(45 183 208)(46 184 209)(47 185 210)(48 186 211)(49 187 212)(50 188 213)(51 189 214)(52 190 215)(53 191 216)(54 192 217)(55 193 218)(56 194 219)(57 195 220)(58 196 201)(59 197 202)(60 198 203)(61 122 115)(62 123 116)(63 124 117)(64 125 118)(65 126 119)(66 127 120)(67 128 101)(68 129 102)(69 130 103)(70 131 104)(71 132 105)(72 133 106)(73 134 107)(74 135 108)(75 136 109)(76 137 110)(77 138 111)(78 139 112)(79 140 113)(80 121 114)
(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 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)(141 148)(142 147)(143 146)(144 145)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(173 180)(174 179)(175 178)(176 177)(181 184)(182 183)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)(201 214)(202 213)(203 212)(204 211)(205 210)(206 209)(207 208)(215 220)(216 219)(217 218)(221 238)(222 237)(223 236)(224 235)(225 234)(226 233)(227 232)(228 231)(229 230)(239 240)
(1 218 93 110 11 208 83 120)(2 209 94 101 12 219 84 111)(3 220 95 112 13 210 85 102)(4 211 96 103 14 201 86 113)(5 202 97 114 15 212 87 104)(6 213 98 105 16 203 88 115)(7 204 99 116 17 214 89 106)(8 215 100 107 18 205 90 117)(9 206 81 118 19 216 91 108)(10 217 82 109 20 207 92 119)(21 64 238 53 31 74 228 43)(22 75 239 44 32 65 229 54)(23 66 240 55 33 76 230 45)(24 77 221 46 34 67 231 56)(25 68 222 57 35 78 232 47)(26 79 223 48 36 69 233 58)(27 70 224 59 37 80 234 49)(28 61 225 50 38 71 235 60)(29 72 226 41 39 62 236 51)(30 63 227 52 40 73 237 42)(121 149 187 171 131 159 197 161)(122 160 188 162 132 150 198 172)(123 151 189 173 133 141 199 163)(124 142 190 164 134 152 200 174)(125 153 191 175 135 143 181 165)(126 144 192 166 136 154 182 176)(127 155 193 177 137 145 183 167)(128 146 194 168 138 156 184 178)(129 157 195 179 139 147 185 169)(130 148 196 170 140 158 186 180)
G:=sub<Sym(240)| (1,240,155)(2,221,156)(3,222,157)(4,223,158)(5,224,159)(6,225,160)(7,226,141)(8,227,142)(9,228,143)(10,229,144)(11,230,145)(12,231,146)(13,232,147)(14,233,148)(15,234,149)(16,235,150)(17,236,151)(18,237,152)(19,238,153)(20,239,154)(21,165,81)(22,166,82)(23,167,83)(24,168,84)(25,169,85)(26,170,86)(27,171,87)(28,172,88)(29,173,89)(30,174,90)(31,175,91)(32,176,92)(33,177,93)(34,178,94)(35,179,95)(36,180,96)(37,161,97)(38,162,98)(39,163,99)(40,164,100)(41,199,204)(42,200,205)(43,181,206)(44,182,207)(45,183,208)(46,184,209)(47,185,210)(48,186,211)(49,187,212)(50,188,213)(51,189,214)(52,190,215)(53,191,216)(54,192,217)(55,193,218)(56,194,219)(57,195,220)(58,196,201)(59,197,202)(60,198,203)(61,122,115)(62,123,116)(63,124,117)(64,125,118)(65,126,119)(66,127,120)(67,128,101)(68,129,102)(69,130,103)(70,131,104)(71,132,105)(72,133,106)(73,134,107)(74,135,108)(75,136,109)(76,137,110)(77,138,111)(78,139,112)(79,140,113)(80,121,114), (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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,184)(182,183)(185,200)(186,199)(187,198)(188,197)(189,196)(190,195)(191,194)(192,193)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(215,220)(216,219)(217,218)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)(239,240), (1,218,93,110,11,208,83,120)(2,209,94,101,12,219,84,111)(3,220,95,112,13,210,85,102)(4,211,96,103,14,201,86,113)(5,202,97,114,15,212,87,104)(6,213,98,105,16,203,88,115)(7,204,99,116,17,214,89,106)(8,215,100,107,18,205,90,117)(9,206,81,118,19,216,91,108)(10,217,82,109,20,207,92,119)(21,64,238,53,31,74,228,43)(22,75,239,44,32,65,229,54)(23,66,240,55,33,76,230,45)(24,77,221,46,34,67,231,56)(25,68,222,57,35,78,232,47)(26,79,223,48,36,69,233,58)(27,70,224,59,37,80,234,49)(28,61,225,50,38,71,235,60)(29,72,226,41,39,62,236,51)(30,63,227,52,40,73,237,42)(121,149,187,171,131,159,197,161)(122,160,188,162,132,150,198,172)(123,151,189,173,133,141,199,163)(124,142,190,164,134,152,200,174)(125,153,191,175,135,143,181,165)(126,144,192,166,136,154,182,176)(127,155,193,177,137,145,183,167)(128,146,194,168,138,156,184,178)(129,157,195,179,139,147,185,169)(130,148,196,170,140,158,186,180)>;
G:=Group( (1,240,155)(2,221,156)(3,222,157)(4,223,158)(5,224,159)(6,225,160)(7,226,141)(8,227,142)(9,228,143)(10,229,144)(11,230,145)(12,231,146)(13,232,147)(14,233,148)(15,234,149)(16,235,150)(17,236,151)(18,237,152)(19,238,153)(20,239,154)(21,165,81)(22,166,82)(23,167,83)(24,168,84)(25,169,85)(26,170,86)(27,171,87)(28,172,88)(29,173,89)(30,174,90)(31,175,91)(32,176,92)(33,177,93)(34,178,94)(35,179,95)(36,180,96)(37,161,97)(38,162,98)(39,163,99)(40,164,100)(41,199,204)(42,200,205)(43,181,206)(44,182,207)(45,183,208)(46,184,209)(47,185,210)(48,186,211)(49,187,212)(50,188,213)(51,189,214)(52,190,215)(53,191,216)(54,192,217)(55,193,218)(56,194,219)(57,195,220)(58,196,201)(59,197,202)(60,198,203)(61,122,115)(62,123,116)(63,124,117)(64,125,118)(65,126,119)(66,127,120)(67,128,101)(68,129,102)(69,130,103)(70,131,104)(71,132,105)(72,133,106)(73,134,107)(74,135,108)(75,136,109)(76,137,110)(77,138,111)(78,139,112)(79,140,113)(80,121,114), (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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,184)(182,183)(185,200)(186,199)(187,198)(188,197)(189,196)(190,195)(191,194)(192,193)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)(215,220)(216,219)(217,218)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)(239,240), (1,218,93,110,11,208,83,120)(2,209,94,101,12,219,84,111)(3,220,95,112,13,210,85,102)(4,211,96,103,14,201,86,113)(5,202,97,114,15,212,87,104)(6,213,98,105,16,203,88,115)(7,204,99,116,17,214,89,106)(8,215,100,107,18,205,90,117)(9,206,81,118,19,216,91,108)(10,217,82,109,20,207,92,119)(21,64,238,53,31,74,228,43)(22,75,239,44,32,65,229,54)(23,66,240,55,33,76,230,45)(24,77,221,46,34,67,231,56)(25,68,222,57,35,78,232,47)(26,79,223,48,36,69,233,58)(27,70,224,59,37,80,234,49)(28,61,225,50,38,71,235,60)(29,72,226,41,39,62,236,51)(30,63,227,52,40,73,237,42)(121,149,187,171,131,159,197,161)(122,160,188,162,132,150,198,172)(123,151,189,173,133,141,199,163)(124,142,190,164,134,152,200,174)(125,153,191,175,135,143,181,165)(126,144,192,166,136,154,182,176)(127,155,193,177,137,145,183,167)(128,146,194,168,138,156,184,178)(129,157,195,179,139,147,185,169)(130,148,196,170,140,158,186,180) );
G=PermutationGroup([[(1,240,155),(2,221,156),(3,222,157),(4,223,158),(5,224,159),(6,225,160),(7,226,141),(8,227,142),(9,228,143),(10,229,144),(11,230,145),(12,231,146),(13,232,147),(14,233,148),(15,234,149),(16,235,150),(17,236,151),(18,237,152),(19,238,153),(20,239,154),(21,165,81),(22,166,82),(23,167,83),(24,168,84),(25,169,85),(26,170,86),(27,171,87),(28,172,88),(29,173,89),(30,174,90),(31,175,91),(32,176,92),(33,177,93),(34,178,94),(35,179,95),(36,180,96),(37,161,97),(38,162,98),(39,163,99),(40,164,100),(41,199,204),(42,200,205),(43,181,206),(44,182,207),(45,183,208),(46,184,209),(47,185,210),(48,186,211),(49,187,212),(50,188,213),(51,189,214),(52,190,215),(53,191,216),(54,192,217),(55,193,218),(56,194,219),(57,195,220),(58,196,201),(59,197,202),(60,198,203),(61,122,115),(62,123,116),(63,124,117),(64,125,118),(65,126,119),(66,127,120),(67,128,101),(68,129,102),(69,130,103),(70,131,104),(71,132,105),(72,133,106),(73,134,107),(74,135,108),(75,136,109),(76,137,110),(77,138,111),(78,139,112),(79,140,113),(80,121,114)], [(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),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,118),(102,117),(103,116),(104,115),(105,114),(106,113),(107,112),(108,111),(109,110),(119,120),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(133,140),(134,139),(135,138),(136,137),(141,148),(142,147),(143,146),(144,145),(149,160),(150,159),(151,158),(152,157),(153,156),(154,155),(161,172),(162,171),(163,170),(164,169),(165,168),(166,167),(173,180),(174,179),(175,178),(176,177),(181,184),(182,183),(185,200),(186,199),(187,198),(188,197),(189,196),(190,195),(191,194),(192,193),(201,214),(202,213),(203,212),(204,211),(205,210),(206,209),(207,208),(215,220),(216,219),(217,218),(221,238),(222,237),(223,236),(224,235),(225,234),(226,233),(227,232),(228,231),(229,230),(239,240)], [(1,218,93,110,11,208,83,120),(2,209,94,101,12,219,84,111),(3,220,95,112,13,210,85,102),(4,211,96,103,14,201,86,113),(5,202,97,114,15,212,87,104),(6,213,98,105,16,203,88,115),(7,204,99,116,17,214,89,106),(8,215,100,107,18,205,90,117),(9,206,81,118,19,216,91,108),(10,217,82,109,20,207,92,119),(21,64,238,53,31,74,228,43),(22,75,239,44,32,65,229,54),(23,66,240,55,33,76,230,45),(24,77,221,46,34,67,231,56),(25,68,222,57,35,78,232,47),(26,79,223,48,36,69,233,58),(27,70,224,59,37,80,234,49),(28,61,225,50,38,71,235,60),(29,72,226,41,39,62,236,51),(30,63,227,52,40,73,237,42),(121,149,187,171,131,159,197,161),(122,160,188,162,132,150,198,172),(123,151,189,173,133,141,199,163),(124,142,190,164,134,152,200,174),(125,153,191,175,135,143,181,165),(126,144,192,166,136,154,182,176),(127,155,193,177,137,145,183,167),(128,146,194,168,138,156,184,178),(129,157,195,179,139,147,185,169),(130,148,196,170,140,158,186,180)]])
120 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 | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | ··· | 24H | 24I | ··· | 24P | 24Q | 24R | 24S | 24T | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| 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 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 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 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
120 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 | 4 | 4 |
| 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 | D20.2C4 | C3×D20.2C4 |
| kernel | C3×D20.2C4 | D5×C24 | C3×C8⋊D5 | C6×C5⋊2C8 | C15×M4(2) | C3×C4○D20 | D20.2C4 | C3×Dic10 | C3×D20 | C3×C5⋊D4 | C8×D5 | C8⋊D5 | C2×C5⋊2C8 | C5×M4(2) | C4○D20 | Dic10 | D20 | C5⋊D4 | C3×M4(2) | C24 | C2×C12 | M4(2) | 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 | 4 | 8 |
Matrix representation of C3×D20.2C4 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 49 | 164 |
| 0 | 0 | 147 | 192 |
| 52 | 1 | 0 | 0 |
| 189 | 189 | 0 | 0 |
| 0 | 0 | 49 | 164 |
| 0 | 0 | 3 | 192 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 82 | 211 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[52,240,0,0,1,0,0,0,0,0,49,147,0,0,164,192],[52,189,0,0,1,189,0,0,0,0,49,3,0,0,164,192],[64,0,0,0,0,64,0,0,0,0,30,82,0,0,0,211] >;
C3×D20.2C4 in GAP, Magma, Sage, TeX
C_3\times D_{20}._2C_4 % in TeX
G:=Group("C3xD20.2C4"); // GroupNames label
G:=SmallGroup(480,700);
// by ID
G=gap.SmallGroup(480,700);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,555,142,102,18822]);
// 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,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations