metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D5×C12)⋊3C4, (C4×Dic3)⋊9D5, (C4×D5)⋊3Dic3, C12.80(C4×D5), C60.153(C2×C4), (Dic3×C20)⋊8C2, (C2×C20).333D6, C3⋊5(C42⋊D5), C4.24(D5×Dic3), C15⋊8(C42⋊C2), (C4×Dic15)⋊30C2, C30.24(C4○D4), C6.23(C4○D20), (C2×C12).337D10, (C2×C30).47C23, C30.Q8⋊42C2, C20.50(C2×Dic3), (C22×D5).81D6, C10.26(C4○D12), C30.119(C22×C4), (C2×C60).235C22, D10.10(C2×Dic3), (C2×Dic5).159D6, C5⋊2(C23.26D6), (C2×Dic3).137D10, Dic5.13(C2×Dic3), D10⋊Dic3.15C2, C2.1(D6.D10), C10.21(C22×Dic3), (C6×Dic5).180C22, (C10×Dic3).161C22, (C2×Dic15).185C22, C6.84(C2×C4×D5), (C2×C4×D5).12S3, (D5×C2×C12).10C2, C2.10(C2×D5×Dic3), C22.32(C2×S3×D5), (C6×D5).50(C2×C4), (C2×C4).238(S3×D5), (D5×C2×C6).94C22, (C2×C6).59(C22×D5), (C2×C10).59(C22×S3), (C3×Dic5).58(C2×C4), SmallGroup(480,433)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D5×C12)⋊C4
G = < a,b,c,d | a12=b5=c2=d4=1, ab=ba, ac=ca, dad-1=a5, cbc=b-1, bd=db, dcd-1=a6c >
Subgroups: 604 in 152 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C23.26D6, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C42⋊D5, D10⋊Dic3, C30.Q8, Dic3×C20, C4×Dic15, D5×C2×C12, (D5×C12)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C4○D4, D10, C2×Dic3, C22×S3, C42⋊C2, C4×D5, C22×D5, C4○D12, C22×Dic3, S3×D5, C2×C4×D5, C4○D20, C23.26D6, D5×Dic3, C2×S3×D5, C42⋊D5, D6.D10, C2×D5×Dic3, (D5×C12)⋊C4
►On 240 points
Generators in S240
(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)
(1 162 133 43 52)(2 163 134 44 53)(3 164 135 45 54)(4 165 136 46 55)(5 166 137 47 56)(6 167 138 48 57)(7 168 139 37 58)(8 157 140 38 59)(9 158 141 39 60)(10 159 142 40 49)(11 160 143 41 50)(12 161 144 42 51)(13 145 209 116 78)(14 146 210 117 79)(15 147 211 118 80)(16 148 212 119 81)(17 149 213 120 82)(18 150 214 109 83)(19 151 215 110 84)(20 152 216 111 73)(21 153 205 112 74)(22 154 206 113 75)(23 155 207 114 76)(24 156 208 115 77)(25 96 217 130 236)(26 85 218 131 237)(27 86 219 132 238)(28 87 220 121 239)(29 88 221 122 240)(30 89 222 123 229)(31 90 223 124 230)(32 91 224 125 231)(33 92 225 126 232)(34 93 226 127 233)(35 94 227 128 234)(36 95 228 129 235)(61 190 176 198 107)(62 191 177 199 108)(63 192 178 200 97)(64 181 179 201 98)(65 182 180 202 99)(66 183 169 203 100)(67 184 170 204 101)(68 185 171 193 102)(69 186 172 194 103)(70 187 173 195 104)(71 188 174 196 105)(72 189 175 197 106)
(1 210)(2 211)(3 212)(4 213)(5 214)(6 215)(7 216)(8 205)(9 206)(10 207)(11 208)(12 209)(13 144)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 97)(36 98)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 109)(57 110)(58 111)(59 112)(60 113)(61 232)(62 233)(63 234)(64 235)(65 236)(66 237)(67 238)(68 239)(69 240)(70 229)(71 230)(72 231)(85 203)(86 204)(87 193)(88 194)(89 195)(90 196)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 181)(130 182)(131 183)(132 184)(145 161)(146 162)(147 163)(148 164)(149 165)(150 166)(151 167)(152 168)(153 157)(154 158)(155 159)(156 160)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 225)(177 226)(178 227)(179 228)(180 217)
(1 232 117 184)(2 237 118 189)(3 230 119 182)(4 235 120 187)(5 240 109 192)(6 233 110 185)(7 238 111 190)(8 231 112 183)(9 236 113 188)(10 229 114 181)(11 234 115 186)(12 239 116 191)(13 199 144 87)(14 204 133 92)(15 197 134 85)(16 202 135 90)(17 195 136 95)(18 200 137 88)(19 193 138 93)(20 198 139 86)(21 203 140 91)(22 196 141 96)(23 201 142 89)(24 194 143 94)(25 75 174 158)(26 80 175 163)(27 73 176 168)(28 78 177 161)(29 83 178 166)(30 76 179 159)(31 81 180 164)(32 74 169 157)(33 79 170 162)(34 84 171 167)(35 77 172 160)(36 82 173 165)(37 219 152 107)(38 224 153 100)(39 217 154 105)(40 222 155 98)(41 227 156 103)(42 220 145 108)(43 225 146 101)(44 218 147 106)(45 223 148 99)(46 228 149 104)(47 221 150 97)(48 226 151 102)(49 123 207 64)(50 128 208 69)(51 121 209 62)(52 126 210 67)(53 131 211 72)(54 124 212 65)(55 129 213 70)(56 122 214 63)(57 127 215 68)(58 132 216 61)(59 125 205 66)(60 130 206 71)
G:=sub<Sym(240)| (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), (1,162,133,43,52)(2,163,134,44,53)(3,164,135,45,54)(4,165,136,46,55)(5,166,137,47,56)(6,167,138,48,57)(7,168,139,37,58)(8,157,140,38,59)(9,158,141,39,60)(10,159,142,40,49)(11,160,143,41,50)(12,161,144,42,51)(13,145,209,116,78)(14,146,210,117,79)(15,147,211,118,80)(16,148,212,119,81)(17,149,213,120,82)(18,150,214,109,83)(19,151,215,110,84)(20,152,216,111,73)(21,153,205,112,74)(22,154,206,113,75)(23,155,207,114,76)(24,156,208,115,77)(25,96,217,130,236)(26,85,218,131,237)(27,86,219,132,238)(28,87,220,121,239)(29,88,221,122,240)(30,89,222,123,229)(31,90,223,124,230)(32,91,224,125,231)(33,92,225,126,232)(34,93,226,127,233)(35,94,227,128,234)(36,95,228,129,235)(61,190,176,198,107)(62,191,177,199,108)(63,192,178,200,97)(64,181,179,201,98)(65,182,180,202,99)(66,183,169,203,100)(67,184,170,204,101)(68,185,171,193,102)(69,186,172,194,103)(70,187,173,195,104)(71,188,174,196,105)(72,189,175,197,106), (1,210)(2,211)(3,212)(4,213)(5,214)(6,215)(7,216)(8,205)(9,206)(10,207)(11,208)(12,209)(13,144)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,109)(57,110)(58,111)(59,112)(60,113)(61,232)(62,233)(63,234)(64,235)(65,236)(66,237)(67,238)(68,239)(69,240)(70,229)(71,230)(72,231)(85,203)(86,204)(87,193)(88,194)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,181)(130,182)(131,183)(132,184)(145,161)(146,162)(147,163)(148,164)(149,165)(150,166)(151,167)(152,168)(153,157)(154,158)(155,159)(156,160)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,225)(177,226)(178,227)(179,228)(180,217), (1,232,117,184)(2,237,118,189)(3,230,119,182)(4,235,120,187)(5,240,109,192)(6,233,110,185)(7,238,111,190)(8,231,112,183)(9,236,113,188)(10,229,114,181)(11,234,115,186)(12,239,116,191)(13,199,144,87)(14,204,133,92)(15,197,134,85)(16,202,135,90)(17,195,136,95)(18,200,137,88)(19,193,138,93)(20,198,139,86)(21,203,140,91)(22,196,141,96)(23,201,142,89)(24,194,143,94)(25,75,174,158)(26,80,175,163)(27,73,176,168)(28,78,177,161)(29,83,178,166)(30,76,179,159)(31,81,180,164)(32,74,169,157)(33,79,170,162)(34,84,171,167)(35,77,172,160)(36,82,173,165)(37,219,152,107)(38,224,153,100)(39,217,154,105)(40,222,155,98)(41,227,156,103)(42,220,145,108)(43,225,146,101)(44,218,147,106)(45,223,148,99)(46,228,149,104)(47,221,150,97)(48,226,151,102)(49,123,207,64)(50,128,208,69)(51,121,209,62)(52,126,210,67)(53,131,211,72)(54,124,212,65)(55,129,213,70)(56,122,214,63)(57,127,215,68)(58,132,216,61)(59,125,205,66)(60,130,206,71)>;
G:=Group( (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), (1,162,133,43,52)(2,163,134,44,53)(3,164,135,45,54)(4,165,136,46,55)(5,166,137,47,56)(6,167,138,48,57)(7,168,139,37,58)(8,157,140,38,59)(9,158,141,39,60)(10,159,142,40,49)(11,160,143,41,50)(12,161,144,42,51)(13,145,209,116,78)(14,146,210,117,79)(15,147,211,118,80)(16,148,212,119,81)(17,149,213,120,82)(18,150,214,109,83)(19,151,215,110,84)(20,152,216,111,73)(21,153,205,112,74)(22,154,206,113,75)(23,155,207,114,76)(24,156,208,115,77)(25,96,217,130,236)(26,85,218,131,237)(27,86,219,132,238)(28,87,220,121,239)(29,88,221,122,240)(30,89,222,123,229)(31,90,223,124,230)(32,91,224,125,231)(33,92,225,126,232)(34,93,226,127,233)(35,94,227,128,234)(36,95,228,129,235)(61,190,176,198,107)(62,191,177,199,108)(63,192,178,200,97)(64,181,179,201,98)(65,182,180,202,99)(66,183,169,203,100)(67,184,170,204,101)(68,185,171,193,102)(69,186,172,194,103)(70,187,173,195,104)(71,188,174,196,105)(72,189,175,197,106), (1,210)(2,211)(3,212)(4,213)(5,214)(6,215)(7,216)(8,205)(9,206)(10,207)(11,208)(12,209)(13,144)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,109)(57,110)(58,111)(59,112)(60,113)(61,232)(62,233)(63,234)(64,235)(65,236)(66,237)(67,238)(68,239)(69,240)(70,229)(71,230)(72,231)(85,203)(86,204)(87,193)(88,194)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,181)(130,182)(131,183)(132,184)(145,161)(146,162)(147,163)(148,164)(149,165)(150,166)(151,167)(152,168)(153,157)(154,158)(155,159)(156,160)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,225)(177,226)(178,227)(179,228)(180,217), (1,232,117,184)(2,237,118,189)(3,230,119,182)(4,235,120,187)(5,240,109,192)(6,233,110,185)(7,238,111,190)(8,231,112,183)(9,236,113,188)(10,229,114,181)(11,234,115,186)(12,239,116,191)(13,199,144,87)(14,204,133,92)(15,197,134,85)(16,202,135,90)(17,195,136,95)(18,200,137,88)(19,193,138,93)(20,198,139,86)(21,203,140,91)(22,196,141,96)(23,201,142,89)(24,194,143,94)(25,75,174,158)(26,80,175,163)(27,73,176,168)(28,78,177,161)(29,83,178,166)(30,76,179,159)(31,81,180,164)(32,74,169,157)(33,79,170,162)(34,84,171,167)(35,77,172,160)(36,82,173,165)(37,219,152,107)(38,224,153,100)(39,217,154,105)(40,222,155,98)(41,227,156,103)(42,220,145,108)(43,225,146,101)(44,218,147,106)(45,223,148,99)(46,228,149,104)(47,221,150,97)(48,226,151,102)(49,123,207,64)(50,128,208,69)(51,121,209,62)(52,126,210,67)(53,131,211,72)(54,124,212,65)(55,129,213,70)(56,122,214,63)(57,127,215,68)(58,132,216,61)(59,125,205,66)(60,130,206,71) );
G=PermutationGroup([[(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)], [(1,162,133,43,52),(2,163,134,44,53),(3,164,135,45,54),(4,165,136,46,55),(5,166,137,47,56),(6,167,138,48,57),(7,168,139,37,58),(8,157,140,38,59),(9,158,141,39,60),(10,159,142,40,49),(11,160,143,41,50),(12,161,144,42,51),(13,145,209,116,78),(14,146,210,117,79),(15,147,211,118,80),(16,148,212,119,81),(17,149,213,120,82),(18,150,214,109,83),(19,151,215,110,84),(20,152,216,111,73),(21,153,205,112,74),(22,154,206,113,75),(23,155,207,114,76),(24,156,208,115,77),(25,96,217,130,236),(26,85,218,131,237),(27,86,219,132,238),(28,87,220,121,239),(29,88,221,122,240),(30,89,222,123,229),(31,90,223,124,230),(32,91,224,125,231),(33,92,225,126,232),(34,93,226,127,233),(35,94,227,128,234),(36,95,228,129,235),(61,190,176,198,107),(62,191,177,199,108),(63,192,178,200,97),(64,181,179,201,98),(65,182,180,202,99),(66,183,169,203,100),(67,184,170,204,101),(68,185,171,193,102),(69,186,172,194,103),(70,187,173,195,104),(71,188,174,196,105),(72,189,175,197,106)], [(1,210),(2,211),(3,212),(4,213),(5,214),(6,215),(7,216),(8,205),(9,206),(10,207),(11,208),(12,209),(13,144),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,141),(23,142),(24,143),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,97),(36,98),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,109),(57,110),(58,111),(59,112),(60,113),(61,232),(62,233),(63,234),(64,235),(65,236),(66,237),(67,238),(68,239),(69,240),(70,229),(71,230),(72,231),(85,203),(86,204),(87,193),(88,194),(89,195),(90,196),(91,197),(92,198),(93,199),(94,200),(95,201),(96,202),(121,185),(122,186),(123,187),(124,188),(125,189),(126,190),(127,191),(128,192),(129,181),(130,182),(131,183),(132,184),(145,161),(146,162),(147,163),(148,164),(149,165),(150,166),(151,167),(152,168),(153,157),(154,158),(155,159),(156,160),(169,218),(170,219),(171,220),(172,221),(173,222),(174,223),(175,224),(176,225),(177,226),(178,227),(179,228),(180,217)], [(1,232,117,184),(2,237,118,189),(3,230,119,182),(4,235,120,187),(5,240,109,192),(6,233,110,185),(7,238,111,190),(8,231,112,183),(9,236,113,188),(10,229,114,181),(11,234,115,186),(12,239,116,191),(13,199,144,87),(14,204,133,92),(15,197,134,85),(16,202,135,90),(17,195,136,95),(18,200,137,88),(19,193,138,93),(20,198,139,86),(21,203,140,91),(22,196,141,96),(23,201,142,89),(24,194,143,94),(25,75,174,158),(26,80,175,163),(27,73,176,168),(28,78,177,161),(29,83,178,166),(30,76,179,159),(31,81,180,164),(32,74,169,157),(33,79,170,162),(34,84,171,167),(35,77,172,160),(36,82,173,165),(37,219,152,107),(38,224,153,100),(39,217,154,105),(40,222,155,98),(41,227,156,103),(42,220,145,108),(43,225,146,101),(44,218,147,106),(45,223,148,99),(46,228,149,104),(47,221,150,97),(48,226,151,102),(49,123,207,64),(50,128,208,69),(51,121,209,62),(52,126,210,67),(53,131,211,72),(54,124,212,65),(55,129,213,70),(56,122,214,63),(57,127,215,68),(58,132,216,61),(59,125,205,66),(60,130,206,71)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 3 | 4 | 4 | 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 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4×D5 | C4○D12 | C4○D20 | S3×D5 | D5×Dic3 | C2×S3×D5 | D6.D10 |
| kernel | (D5×C12)⋊C4 | D10⋊Dic3 | C30.Q8 | Dic3×C20 | C4×Dic15 | D5×C2×C12 | D5×C12 | C2×C4×D5 | C4×Dic3 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C12 | C10 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 8 | 8 | 16 | 2 | 4 | 2 | 8 |
Matrix representation of (D5×C12)⋊C4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 29 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 43 | 60 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 53 | 1 | 0 | 0 |
| 0 | 0 | 0 | 43 | 60 |
| 0 | 0 | 0 | 18 | 18 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 41 | 5 | 0 | 0 |
| 0 | 54 | 20 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,32,29,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,43,1,0,0,0,60,0],[1,0,0,0,0,0,60,53,0,0,0,0,1,0,0,0,0,0,43,18,0,0,0,60,18],[50,0,0,0,0,0,41,54,0,0,0,5,20,0,0,0,0,0,60,0,0,0,0,0,60] >;
(D5×C12)⋊C4 in GAP, Magma, Sage, TeX
(D_5\times C_{12})\rtimes C_4 % in TeX
G:=Group("(D5xC12):C4"); // GroupNames label
G:=SmallGroup(480,433);
// by ID
G=gap.SmallGroup(480,433);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^5=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^5,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^6*c>;
// generators/relations