direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.53D4, C60.238D4, C3⋊C8.1C20, C4.13(S3×C20), C12.5(C2×C20), C12.53(C5×D4), C30.60(C4⋊C4), (C2×C30).10Q8, C20.115(C4×S3), C60.175(C2×C4), (C2×C20).348D6, (C2×C10).4Dic6, C15⋊15(C8.C4), (C5×M4(2)).1S3, M4(2).1(C5×S3), C4.Dic3.2C10, C22.1(C5×Dic6), C20.121(C3⋊D4), (C2×C60).341C22, (C15×M4(2)).3C2, (C3×M4(2)).1C10, C10.25(Dic3⋊C4), (C2×C6).(C5×Q8), (C5×C3⋊C8).8C4, C6.8(C5×C4⋊C4), (C2×C3⋊C8).4C10, C3⋊2(C5×C8.C4), (C10×C3⋊C8).16C2, C4.28(C5×C3⋊D4), (C2×C4).36(S3×C10), C2.5(C5×Dic3⋊C4), (C2×C12).11(C2×C10), (C5×C4.Dic3).6C2, SmallGroup(480,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.53D4
G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c3 >
Subgroups: 100 in 60 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), M4(2), C20, C2×C10, C3⋊C8, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60, C2×C30, C2×C40, C5×M4(2), C5×M4(2), C12.53D4, C5×C3⋊C8, C5×C3⋊C8, C120, C2×C60, C5×C8.C4, C10×C3⋊C8, C5×C4.Dic3, C15×M4(2), C5×C12.53D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, D6, C4⋊C4, C20, C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C8.C4, C2×C20, C5×D4, C5×Q8, Dic3⋊C4, S3×C10, C5×C4⋊C4, C12.53D4, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C8.C4, C5×Dic3⋊C4, C5×C12.53D4
►On 240 points
Generators in S240
(1 58 46 25 19)(2 59 47 26 20)(3 60 48 27 21)(4 49 37 28 22)(5 50 38 29 23)(6 51 39 30 24)(7 52 40 31 13)(8 53 41 32 14)(9 54 42 33 15)(10 55 43 34 16)(11 56 44 35 17)(12 57 45 36 18)(61 115 100 94 82)(62 116 101 95 83)(63 117 102 96 84)(64 118 103 85 73)(65 119 104 86 74)(66 120 105 87 75)(67 109 106 88 76)(68 110 107 89 77)(69 111 108 90 78)(70 112 97 91 79)(71 113 98 92 80)(72 114 99 93 81)(121 174 162 153 137)(122 175 163 154 138)(123 176 164 155 139)(124 177 165 156 140)(125 178 166 145 141)(126 179 167 146 142)(127 180 168 147 143)(128 169 157 148 144)(129 170 158 149 133)(130 171 159 150 134)(131 172 160 151 135)(132 173 161 152 136)(181 234 227 205 203)(182 235 228 206 204)(183 236 217 207 193)(184 237 218 208 194)(185 238 219 209 195)(186 239 220 210 196)(187 240 221 211 197)(188 229 222 212 198)(189 230 223 213 199)(190 231 224 214 200)(191 232 225 215 201)(192 233 226 216 202)
(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)
(1 126 4 129 7 132 10 123)(2 131 5 122 8 125 11 128)(3 124 6 127 9 130 12 121)(13 136 16 139 19 142 22 133)(14 141 17 144 20 135 23 138)(15 134 18 137 21 140 24 143)(25 146 28 149 31 152 34 155)(26 151 29 154 32 145 35 148)(27 156 30 147 33 150 36 153)(37 158 40 161 43 164 46 167)(38 163 41 166 44 157 47 160)(39 168 42 159 45 162 48 165)(49 170 52 173 55 176 58 179)(50 175 53 178 56 169 59 172)(51 180 54 171 57 174 60 177)(61 190 70 187 67 184 64 181)(62 183 71 192 68 189 65 186)(63 188 72 185 69 182 66 191)(73 203 82 200 79 197 76 194)(74 196 83 193 80 202 77 199)(75 201 84 198 81 195 78 204)(85 205 94 214 91 211 88 208)(86 210 95 207 92 216 89 213)(87 215 96 212 93 209 90 206)(97 221 106 218 103 227 100 224)(98 226 107 223 104 220 101 217)(99 219 108 228 105 225 102 222)(109 237 118 234 115 231 112 240)(110 230 119 239 116 236 113 233)(111 235 120 232 117 229 114 238)
(1 187 10 184 7 181 4 190)(2 192 11 189 8 186 5 183)(3 185 12 182 9 191 6 188)(13 203 22 200 19 197 16 194)(14 196 23 193 20 202 17 199)(15 201 24 198 21 195 18 204)(25 211 34 208 31 205 28 214)(26 216 35 213 32 210 29 207)(27 209 36 206 33 215 30 212)(37 224 46 221 43 218 40 227)(38 217 47 226 44 223 41 220)(39 222 48 219 45 228 42 225)(49 231 58 240 55 237 52 234)(50 236 59 233 56 230 53 239)(51 229 60 238 57 235 54 232)(61 126 70 123 67 132 64 129)(62 131 71 128 68 125 65 122)(63 124 72 121 69 130 66 127)(73 133 82 142 79 139 76 136)(74 138 83 135 80 144 77 141)(75 143 84 140 81 137 78 134)(85 149 94 146 91 155 88 152)(86 154 95 151 92 148 89 145)(87 147 96 156 93 153 90 150)(97 164 106 161 103 158 100 167)(98 157 107 166 104 163 101 160)(99 162 108 159 105 168 102 165)(109 173 118 170 115 179 112 176)(110 178 119 175 116 172 113 169)(111 171 120 180 117 177 114 174)
G:=sub<Sym(240)| (1,58,46,25,19)(2,59,47,26,20)(3,60,48,27,21)(4,49,37,28,22)(5,50,38,29,23)(6,51,39,30,24)(7,52,40,31,13)(8,53,41,32,14)(9,54,42,33,15)(10,55,43,34,16)(11,56,44,35,17)(12,57,45,36,18)(61,115,100,94,82)(62,116,101,95,83)(63,117,102,96,84)(64,118,103,85,73)(65,119,104,86,74)(66,120,105,87,75)(67,109,106,88,76)(68,110,107,89,77)(69,111,108,90,78)(70,112,97,91,79)(71,113,98,92,80)(72,114,99,93,81)(121,174,162,153,137)(122,175,163,154,138)(123,176,164,155,139)(124,177,165,156,140)(125,178,166,145,141)(126,179,167,146,142)(127,180,168,147,143)(128,169,157,148,144)(129,170,158,149,133)(130,171,159,150,134)(131,172,160,151,135)(132,173,161,152,136)(181,234,227,205,203)(182,235,228,206,204)(183,236,217,207,193)(184,237,218,208,194)(185,238,219,209,195)(186,239,220,210,196)(187,240,221,211,197)(188,229,222,212,198)(189,230,223,213,199)(190,231,224,214,200)(191,232,225,215,201)(192,233,226,216,202), (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), (1,126,4,129,7,132,10,123)(2,131,5,122,8,125,11,128)(3,124,6,127,9,130,12,121)(13,136,16,139,19,142,22,133)(14,141,17,144,20,135,23,138)(15,134,18,137,21,140,24,143)(25,146,28,149,31,152,34,155)(26,151,29,154,32,145,35,148)(27,156,30,147,33,150,36,153)(37,158,40,161,43,164,46,167)(38,163,41,166,44,157,47,160)(39,168,42,159,45,162,48,165)(49,170,52,173,55,176,58,179)(50,175,53,178,56,169,59,172)(51,180,54,171,57,174,60,177)(61,190,70,187,67,184,64,181)(62,183,71,192,68,189,65,186)(63,188,72,185,69,182,66,191)(73,203,82,200,79,197,76,194)(74,196,83,193,80,202,77,199)(75,201,84,198,81,195,78,204)(85,205,94,214,91,211,88,208)(86,210,95,207,92,216,89,213)(87,215,96,212,93,209,90,206)(97,221,106,218,103,227,100,224)(98,226,107,223,104,220,101,217)(99,219,108,228,105,225,102,222)(109,237,118,234,115,231,112,240)(110,230,119,239,116,236,113,233)(111,235,120,232,117,229,114,238), (1,187,10,184,7,181,4,190)(2,192,11,189,8,186,5,183)(3,185,12,182,9,191,6,188)(13,203,22,200,19,197,16,194)(14,196,23,193,20,202,17,199)(15,201,24,198,21,195,18,204)(25,211,34,208,31,205,28,214)(26,216,35,213,32,210,29,207)(27,209,36,206,33,215,30,212)(37,224,46,221,43,218,40,227)(38,217,47,226,44,223,41,220)(39,222,48,219,45,228,42,225)(49,231,58,240,55,237,52,234)(50,236,59,233,56,230,53,239)(51,229,60,238,57,235,54,232)(61,126,70,123,67,132,64,129)(62,131,71,128,68,125,65,122)(63,124,72,121,69,130,66,127)(73,133,82,142,79,139,76,136)(74,138,83,135,80,144,77,141)(75,143,84,140,81,137,78,134)(85,149,94,146,91,155,88,152)(86,154,95,151,92,148,89,145)(87,147,96,156,93,153,90,150)(97,164,106,161,103,158,100,167)(98,157,107,166,104,163,101,160)(99,162,108,159,105,168,102,165)(109,173,118,170,115,179,112,176)(110,178,119,175,116,172,113,169)(111,171,120,180,117,177,114,174)>;
G:=Group( (1,58,46,25,19)(2,59,47,26,20)(3,60,48,27,21)(4,49,37,28,22)(5,50,38,29,23)(6,51,39,30,24)(7,52,40,31,13)(8,53,41,32,14)(9,54,42,33,15)(10,55,43,34,16)(11,56,44,35,17)(12,57,45,36,18)(61,115,100,94,82)(62,116,101,95,83)(63,117,102,96,84)(64,118,103,85,73)(65,119,104,86,74)(66,120,105,87,75)(67,109,106,88,76)(68,110,107,89,77)(69,111,108,90,78)(70,112,97,91,79)(71,113,98,92,80)(72,114,99,93,81)(121,174,162,153,137)(122,175,163,154,138)(123,176,164,155,139)(124,177,165,156,140)(125,178,166,145,141)(126,179,167,146,142)(127,180,168,147,143)(128,169,157,148,144)(129,170,158,149,133)(130,171,159,150,134)(131,172,160,151,135)(132,173,161,152,136)(181,234,227,205,203)(182,235,228,206,204)(183,236,217,207,193)(184,237,218,208,194)(185,238,219,209,195)(186,239,220,210,196)(187,240,221,211,197)(188,229,222,212,198)(189,230,223,213,199)(190,231,224,214,200)(191,232,225,215,201)(192,233,226,216,202), (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), (1,126,4,129,7,132,10,123)(2,131,5,122,8,125,11,128)(3,124,6,127,9,130,12,121)(13,136,16,139,19,142,22,133)(14,141,17,144,20,135,23,138)(15,134,18,137,21,140,24,143)(25,146,28,149,31,152,34,155)(26,151,29,154,32,145,35,148)(27,156,30,147,33,150,36,153)(37,158,40,161,43,164,46,167)(38,163,41,166,44,157,47,160)(39,168,42,159,45,162,48,165)(49,170,52,173,55,176,58,179)(50,175,53,178,56,169,59,172)(51,180,54,171,57,174,60,177)(61,190,70,187,67,184,64,181)(62,183,71,192,68,189,65,186)(63,188,72,185,69,182,66,191)(73,203,82,200,79,197,76,194)(74,196,83,193,80,202,77,199)(75,201,84,198,81,195,78,204)(85,205,94,214,91,211,88,208)(86,210,95,207,92,216,89,213)(87,215,96,212,93,209,90,206)(97,221,106,218,103,227,100,224)(98,226,107,223,104,220,101,217)(99,219,108,228,105,225,102,222)(109,237,118,234,115,231,112,240)(110,230,119,239,116,236,113,233)(111,235,120,232,117,229,114,238), (1,187,10,184,7,181,4,190)(2,192,11,189,8,186,5,183)(3,185,12,182,9,191,6,188)(13,203,22,200,19,197,16,194)(14,196,23,193,20,202,17,199)(15,201,24,198,21,195,18,204)(25,211,34,208,31,205,28,214)(26,216,35,213,32,210,29,207)(27,209,36,206,33,215,30,212)(37,224,46,221,43,218,40,227)(38,217,47,226,44,223,41,220)(39,222,48,219,45,228,42,225)(49,231,58,240,55,237,52,234)(50,236,59,233,56,230,53,239)(51,229,60,238,57,235,54,232)(61,126,70,123,67,132,64,129)(62,131,71,128,68,125,65,122)(63,124,72,121,69,130,66,127)(73,133,82,142,79,139,76,136)(74,138,83,135,80,144,77,141)(75,143,84,140,81,137,78,134)(85,149,94,146,91,155,88,152)(86,154,95,151,92,148,89,145)(87,147,96,156,93,153,90,150)(97,164,106,161,103,158,100,167)(98,157,107,166,104,163,101,160)(99,162,108,159,105,168,102,165)(109,173,118,170,115,179,112,176)(110,178,119,175,116,172,113,169)(111,171,120,180,117,177,114,174) );
G=PermutationGroup([[(1,58,46,25,19),(2,59,47,26,20),(3,60,48,27,21),(4,49,37,28,22),(5,50,38,29,23),(6,51,39,30,24),(7,52,40,31,13),(8,53,41,32,14),(9,54,42,33,15),(10,55,43,34,16),(11,56,44,35,17),(12,57,45,36,18),(61,115,100,94,82),(62,116,101,95,83),(63,117,102,96,84),(64,118,103,85,73),(65,119,104,86,74),(66,120,105,87,75),(67,109,106,88,76),(68,110,107,89,77),(69,111,108,90,78),(70,112,97,91,79),(71,113,98,92,80),(72,114,99,93,81),(121,174,162,153,137),(122,175,163,154,138),(123,176,164,155,139),(124,177,165,156,140),(125,178,166,145,141),(126,179,167,146,142),(127,180,168,147,143),(128,169,157,148,144),(129,170,158,149,133),(130,171,159,150,134),(131,172,160,151,135),(132,173,161,152,136),(181,234,227,205,203),(182,235,228,206,204),(183,236,217,207,193),(184,237,218,208,194),(185,238,219,209,195),(186,239,220,210,196),(187,240,221,211,197),(188,229,222,212,198),(189,230,223,213,199),(190,231,224,214,200),(191,232,225,215,201),(192,233,226,216,202)], [(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)], [(1,126,4,129,7,132,10,123),(2,131,5,122,8,125,11,128),(3,124,6,127,9,130,12,121),(13,136,16,139,19,142,22,133),(14,141,17,144,20,135,23,138),(15,134,18,137,21,140,24,143),(25,146,28,149,31,152,34,155),(26,151,29,154,32,145,35,148),(27,156,30,147,33,150,36,153),(37,158,40,161,43,164,46,167),(38,163,41,166,44,157,47,160),(39,168,42,159,45,162,48,165),(49,170,52,173,55,176,58,179),(50,175,53,178,56,169,59,172),(51,180,54,171,57,174,60,177),(61,190,70,187,67,184,64,181),(62,183,71,192,68,189,65,186),(63,188,72,185,69,182,66,191),(73,203,82,200,79,197,76,194),(74,196,83,193,80,202,77,199),(75,201,84,198,81,195,78,204),(85,205,94,214,91,211,88,208),(86,210,95,207,92,216,89,213),(87,215,96,212,93,209,90,206),(97,221,106,218,103,227,100,224),(98,226,107,223,104,220,101,217),(99,219,108,228,105,225,102,222),(109,237,118,234,115,231,112,240),(110,230,119,239,116,236,113,233),(111,235,120,232,117,229,114,238)], [(1,187,10,184,7,181,4,190),(2,192,11,189,8,186,5,183),(3,185,12,182,9,191,6,188),(13,203,22,200,19,197,16,194),(14,196,23,193,20,202,17,199),(15,201,24,198,21,195,18,204),(25,211,34,208,31,205,28,214),(26,216,35,213,32,210,29,207),(27,209,36,206,33,215,30,212),(37,224,46,221,43,218,40,227),(38,217,47,226,44,223,41,220),(39,222,48,219,45,228,42,225),(49,231,58,240,55,237,52,234),(50,236,59,233,56,230,53,239),(51,229,60,238,57,235,54,232),(61,126,70,123,67,132,64,129),(62,131,71,128,68,125,65,122),(63,124,72,121,69,130,66,127),(73,133,82,142,79,139,76,136),(74,138,83,135,80,144,77,141),(75,143,84,140,81,137,78,134),(85,149,94,146,91,155,88,152),(86,154,95,151,92,148,89,145),(87,147,96,156,93,153,90,150),(97,164,106,161,103,158,100,167),(98,157,107,166,104,163,101,160),(99,162,108,159,105,168,102,165),(109,173,118,170,115,179,112,176),(110,178,119,175,116,172,113,169),(111,171,120,180,117,177,114,174)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 40I | ··· | 40X | 40Y | ··· | 40AF | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
120 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | Q8 | D6 | C4×S3 | C3⋊D4 | Dic6 | C5×S3 | C8.C4 | C5×D4 | C5×Q8 | S3×C10 | S3×C20 | C5×C3⋊D4 | C5×Dic6 | C5×C8.C4 | C12.53D4 | C5×C12.53D4 |
| kernel | C5×C12.53D4 | C10×C3⋊C8 | C5×C4.Dic3 | C15×M4(2) | C5×C3⋊C8 | C12.53D4 | C2×C3⋊C8 | C4.Dic3 | C3×M4(2) | C3⋊C8 | C5×M4(2) | C60 | C2×C30 | C2×C20 | C20 | C20 | C2×C10 | M4(2) | C15 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 2 | 8 |
Matrix representation of C5×C12.53D4 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 149 | 48 | 0 | 0 |
| 140 | 92 | 0 | 0 |
| 0 | 0 | 233 | 0 |
| 0 | 0 | 130 | 30 |
| 92 | 193 | 0 | 0 |
| 101 | 149 | 0 | 0 |
| 0 | 0 | 143 | 90 |
| 0 | 0 | 155 | 98 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[240,1,0,0,240,0,0,0,0,0,177,0,0,0,0,177],[149,140,0,0,48,92,0,0,0,0,233,130,0,0,0,30],[92,101,0,0,193,149,0,0,0,0,143,155,0,0,90,98] >;
C5×C12.53D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}._{53}D_4 % in TeX
G:=Group("C5xC12.53D4"); // GroupNames label
G:=SmallGroup(480,141);
// by ID
G=gap.SmallGroup(480,141);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,589,148,136,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^3>;
// generators/relations