direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.10D4, C60.143D4, (C2×C60).24C4, (C2×C12).1C20, C12.10(C5×D4), (C6×Q8).2C10, (C2×C20).216D6, (Q8×C10).11S3, (Q8×C30).12C2, C20.94(C3⋊D4), (C2×C20).12Dic3, C4.Dic3.4C10, C15⋊12(C4.10D4), (C2×C60).348C22, C22.4(C10×Dic3), C30.121(C22⋊C4), C10.37(C6.D4), (C2×C4).(C5×Dic3), (C2×C4).4(S3×C10), C3⋊2(C5×C4.10D4), C4.15(C5×C3⋊D4), (C2×Q8).4(C5×S3), (C2×C6).30(C2×C20), C6.17(C5×C22⋊C4), (C2×C30).198(C2×C4), (C2×C12).18(C2×C10), C2.7(C5×C6.D4), (C5×C4.Dic3).8C2, (C2×C10).42(C2×Dic3), SmallGroup(480,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.10D4
G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b9c3 >
Subgroups: 132 in 76 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, M4(2), C2×Q8, C20, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C40, C2×C20, C2×C20, C5×Q8, C4.Dic3, C6×Q8, C60, C60, C2×C30, C5×M4(2), Q8×C10, C12.10D4, C5×C3⋊C8, C2×C60, C2×C60, Q8×C15, C5×C4.10D4, C5×C4.Dic3, Q8×C30, C5×C12.10D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C4.10D4, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C12.10D4, C10×Dic3, C5×C3⋊D4, C5×C4.10D4, C5×C6.D4, C5×C12.10D4
►On 240 points
Generators in S240
(1 55 37 34 13)(2 56 38 35 14)(3 57 39 36 15)(4 58 40 25 16)(5 59 41 26 17)(6 60 42 27 18)(7 49 43 28 19)(8 50 44 29 20)(9 51 45 30 21)(10 52 46 31 22)(11 53 47 32 23)(12 54 48 33 24)(61 118 97 94 73)(62 119 98 95 74)(63 120 99 96 75)(64 109 100 85 76)(65 110 101 86 77)(66 111 102 87 78)(67 112 103 88 79)(68 113 104 89 80)(69 114 105 90 81)(70 115 106 91 82)(71 116 107 92 83)(72 117 108 93 84)(121 178 160 148 135)(122 179 161 149 136)(123 180 162 150 137)(124 169 163 151 138)(125 170 164 152 139)(126 171 165 153 140)(127 172 166 154 141)(128 173 167 155 142)(129 174 168 156 143)(130 175 157 145 144)(131 176 158 146 133)(132 177 159 147 134)(181 240 224 215 197)(182 229 225 216 198)(183 230 226 205 199)(184 231 227 206 200)(185 232 228 207 201)(186 233 217 208 202)(187 234 218 209 203)(188 235 219 210 204)(189 236 220 211 193)(190 237 221 212 194)(191 238 222 213 195)(192 239 223 214 196)
(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)
(1 185 4 182 7 191 10 188)(2 184 5 181 8 190 11 187)(3 183 6 192 9 189 12 186)(13 201 16 198 19 195 22 204)(14 200 17 197 20 194 23 203)(15 199 18 196 21 193 24 202)(25 216 28 213 31 210 34 207)(26 215 29 212 32 209 35 206)(27 214 30 211 33 208 36 205)(37 228 40 225 43 222 46 219)(38 227 41 224 44 221 47 218)(39 226 42 223 45 220 48 217)(49 238 52 235 55 232 58 229)(50 237 53 234 56 231 59 240)(51 236 54 233 57 230 60 239)(61 130 64 127 67 124 70 121)(62 129 65 126 68 123 71 132)(63 128 66 125 69 122 72 131)(73 144 76 141 79 138 82 135)(74 143 77 140 80 137 83 134)(75 142 78 139 81 136 84 133)(85 154 88 151 91 148 94 145)(86 153 89 150 92 147 95 156)(87 152 90 149 93 146 96 155)(97 157 100 166 103 163 106 160)(98 168 101 165 104 162 107 159)(99 167 102 164 105 161 108 158)(109 172 112 169 115 178 118 175)(110 171 113 180 116 177 119 174)(111 170 114 179 117 176 120 173)
(1 130 10 127 7 124 4 121)(2 123 11 132 8 129 5 126)(3 128 12 125 9 122 6 131)(13 144 22 141 19 138 16 135)(14 137 23 134 20 143 17 140)(15 142 24 139 21 136 18 133)(25 148 34 145 31 154 28 151)(26 153 35 150 32 147 29 156)(27 146 36 155 33 152 30 149)(37 157 46 166 43 163 40 160)(38 162 47 159 44 168 41 165)(39 167 48 164 45 161 42 158)(49 169 58 178 55 175 52 172)(50 174 59 171 56 180 53 177)(51 179 60 176 57 173 54 170)(61 191 70 188 67 185 64 182)(62 184 71 181 68 190 65 187)(63 189 72 186 69 183 66 192)(73 195 82 204 79 201 76 198)(74 200 83 197 80 194 77 203)(75 193 84 202 81 199 78 196)(85 216 94 213 91 210 88 207)(86 209 95 206 92 215 89 212)(87 214 96 211 93 208 90 205)(97 222 106 219 103 228 100 225)(98 227 107 224 104 221 101 218)(99 220 108 217 105 226 102 223)(109 229 118 238 115 235 112 232)(110 234 119 231 116 240 113 237)(111 239 120 236 117 233 114 230)
G:=sub<Sym(240)| (1,55,37,34,13)(2,56,38,35,14)(3,57,39,36,15)(4,58,40,25,16)(5,59,41,26,17)(6,60,42,27,18)(7,49,43,28,19)(8,50,44,29,20)(9,51,45,30,21)(10,52,46,31,22)(11,53,47,32,23)(12,54,48,33,24)(61,118,97,94,73)(62,119,98,95,74)(63,120,99,96,75)(64,109,100,85,76)(65,110,101,86,77)(66,111,102,87,78)(67,112,103,88,79)(68,113,104,89,80)(69,114,105,90,81)(70,115,106,91,82)(71,116,107,92,83)(72,117,108,93,84)(121,178,160,148,135)(122,179,161,149,136)(123,180,162,150,137)(124,169,163,151,138)(125,170,164,152,139)(126,171,165,153,140)(127,172,166,154,141)(128,173,167,155,142)(129,174,168,156,143)(130,175,157,145,144)(131,176,158,146,133)(132,177,159,147,134)(181,240,224,215,197)(182,229,225,216,198)(183,230,226,205,199)(184,231,227,206,200)(185,232,228,207,201)(186,233,217,208,202)(187,234,218,209,203)(188,235,219,210,204)(189,236,220,211,193)(190,237,221,212,194)(191,238,222,213,195)(192,239,223,214,196), (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), (1,185,4,182,7,191,10,188)(2,184,5,181,8,190,11,187)(3,183,6,192,9,189,12,186)(13,201,16,198,19,195,22,204)(14,200,17,197,20,194,23,203)(15,199,18,196,21,193,24,202)(25,216,28,213,31,210,34,207)(26,215,29,212,32,209,35,206)(27,214,30,211,33,208,36,205)(37,228,40,225,43,222,46,219)(38,227,41,224,44,221,47,218)(39,226,42,223,45,220,48,217)(49,238,52,235,55,232,58,229)(50,237,53,234,56,231,59,240)(51,236,54,233,57,230,60,239)(61,130,64,127,67,124,70,121)(62,129,65,126,68,123,71,132)(63,128,66,125,69,122,72,131)(73,144,76,141,79,138,82,135)(74,143,77,140,80,137,83,134)(75,142,78,139,81,136,84,133)(85,154,88,151,91,148,94,145)(86,153,89,150,92,147,95,156)(87,152,90,149,93,146,96,155)(97,157,100,166,103,163,106,160)(98,168,101,165,104,162,107,159)(99,167,102,164,105,161,108,158)(109,172,112,169,115,178,118,175)(110,171,113,180,116,177,119,174)(111,170,114,179,117,176,120,173), (1,130,10,127,7,124,4,121)(2,123,11,132,8,129,5,126)(3,128,12,125,9,122,6,131)(13,144,22,141,19,138,16,135)(14,137,23,134,20,143,17,140)(15,142,24,139,21,136,18,133)(25,148,34,145,31,154,28,151)(26,153,35,150,32,147,29,156)(27,146,36,155,33,152,30,149)(37,157,46,166,43,163,40,160)(38,162,47,159,44,168,41,165)(39,167,48,164,45,161,42,158)(49,169,58,178,55,175,52,172)(50,174,59,171,56,180,53,177)(51,179,60,176,57,173,54,170)(61,191,70,188,67,185,64,182)(62,184,71,181,68,190,65,187)(63,189,72,186,69,183,66,192)(73,195,82,204,79,201,76,198)(74,200,83,197,80,194,77,203)(75,193,84,202,81,199,78,196)(85,216,94,213,91,210,88,207)(86,209,95,206,92,215,89,212)(87,214,96,211,93,208,90,205)(97,222,106,219,103,228,100,225)(98,227,107,224,104,221,101,218)(99,220,108,217,105,226,102,223)(109,229,118,238,115,235,112,232)(110,234,119,231,116,240,113,237)(111,239,120,236,117,233,114,230)>;
G:=Group( (1,55,37,34,13)(2,56,38,35,14)(3,57,39,36,15)(4,58,40,25,16)(5,59,41,26,17)(6,60,42,27,18)(7,49,43,28,19)(8,50,44,29,20)(9,51,45,30,21)(10,52,46,31,22)(11,53,47,32,23)(12,54,48,33,24)(61,118,97,94,73)(62,119,98,95,74)(63,120,99,96,75)(64,109,100,85,76)(65,110,101,86,77)(66,111,102,87,78)(67,112,103,88,79)(68,113,104,89,80)(69,114,105,90,81)(70,115,106,91,82)(71,116,107,92,83)(72,117,108,93,84)(121,178,160,148,135)(122,179,161,149,136)(123,180,162,150,137)(124,169,163,151,138)(125,170,164,152,139)(126,171,165,153,140)(127,172,166,154,141)(128,173,167,155,142)(129,174,168,156,143)(130,175,157,145,144)(131,176,158,146,133)(132,177,159,147,134)(181,240,224,215,197)(182,229,225,216,198)(183,230,226,205,199)(184,231,227,206,200)(185,232,228,207,201)(186,233,217,208,202)(187,234,218,209,203)(188,235,219,210,204)(189,236,220,211,193)(190,237,221,212,194)(191,238,222,213,195)(192,239,223,214,196), (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), (1,185,4,182,7,191,10,188)(2,184,5,181,8,190,11,187)(3,183,6,192,9,189,12,186)(13,201,16,198,19,195,22,204)(14,200,17,197,20,194,23,203)(15,199,18,196,21,193,24,202)(25,216,28,213,31,210,34,207)(26,215,29,212,32,209,35,206)(27,214,30,211,33,208,36,205)(37,228,40,225,43,222,46,219)(38,227,41,224,44,221,47,218)(39,226,42,223,45,220,48,217)(49,238,52,235,55,232,58,229)(50,237,53,234,56,231,59,240)(51,236,54,233,57,230,60,239)(61,130,64,127,67,124,70,121)(62,129,65,126,68,123,71,132)(63,128,66,125,69,122,72,131)(73,144,76,141,79,138,82,135)(74,143,77,140,80,137,83,134)(75,142,78,139,81,136,84,133)(85,154,88,151,91,148,94,145)(86,153,89,150,92,147,95,156)(87,152,90,149,93,146,96,155)(97,157,100,166,103,163,106,160)(98,168,101,165,104,162,107,159)(99,167,102,164,105,161,108,158)(109,172,112,169,115,178,118,175)(110,171,113,180,116,177,119,174)(111,170,114,179,117,176,120,173), (1,130,10,127,7,124,4,121)(2,123,11,132,8,129,5,126)(3,128,12,125,9,122,6,131)(13,144,22,141,19,138,16,135)(14,137,23,134,20,143,17,140)(15,142,24,139,21,136,18,133)(25,148,34,145,31,154,28,151)(26,153,35,150,32,147,29,156)(27,146,36,155,33,152,30,149)(37,157,46,166,43,163,40,160)(38,162,47,159,44,168,41,165)(39,167,48,164,45,161,42,158)(49,169,58,178,55,175,52,172)(50,174,59,171,56,180,53,177)(51,179,60,176,57,173,54,170)(61,191,70,188,67,185,64,182)(62,184,71,181,68,190,65,187)(63,189,72,186,69,183,66,192)(73,195,82,204,79,201,76,198)(74,200,83,197,80,194,77,203)(75,193,84,202,81,199,78,196)(85,216,94,213,91,210,88,207)(86,209,95,206,92,215,89,212)(87,214,96,211,93,208,90,205)(97,222,106,219,103,228,100,225)(98,227,107,224,104,221,101,218)(99,220,108,217,105,226,102,223)(109,229,118,238,115,235,112,232)(110,234,119,231,116,240,113,237)(111,239,120,236,117,233,114,230) );
G=PermutationGroup([[(1,55,37,34,13),(2,56,38,35,14),(3,57,39,36,15),(4,58,40,25,16),(5,59,41,26,17),(6,60,42,27,18),(7,49,43,28,19),(8,50,44,29,20),(9,51,45,30,21),(10,52,46,31,22),(11,53,47,32,23),(12,54,48,33,24),(61,118,97,94,73),(62,119,98,95,74),(63,120,99,96,75),(64,109,100,85,76),(65,110,101,86,77),(66,111,102,87,78),(67,112,103,88,79),(68,113,104,89,80),(69,114,105,90,81),(70,115,106,91,82),(71,116,107,92,83),(72,117,108,93,84),(121,178,160,148,135),(122,179,161,149,136),(123,180,162,150,137),(124,169,163,151,138),(125,170,164,152,139),(126,171,165,153,140),(127,172,166,154,141),(128,173,167,155,142),(129,174,168,156,143),(130,175,157,145,144),(131,176,158,146,133),(132,177,159,147,134),(181,240,224,215,197),(182,229,225,216,198),(183,230,226,205,199),(184,231,227,206,200),(185,232,228,207,201),(186,233,217,208,202),(187,234,218,209,203),(188,235,219,210,204),(189,236,220,211,193),(190,237,221,212,194),(191,238,222,213,195),(192,239,223,214,196)], [(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)], [(1,185,4,182,7,191,10,188),(2,184,5,181,8,190,11,187),(3,183,6,192,9,189,12,186),(13,201,16,198,19,195,22,204),(14,200,17,197,20,194,23,203),(15,199,18,196,21,193,24,202),(25,216,28,213,31,210,34,207),(26,215,29,212,32,209,35,206),(27,214,30,211,33,208,36,205),(37,228,40,225,43,222,46,219),(38,227,41,224,44,221,47,218),(39,226,42,223,45,220,48,217),(49,238,52,235,55,232,58,229),(50,237,53,234,56,231,59,240),(51,236,54,233,57,230,60,239),(61,130,64,127,67,124,70,121),(62,129,65,126,68,123,71,132),(63,128,66,125,69,122,72,131),(73,144,76,141,79,138,82,135),(74,143,77,140,80,137,83,134),(75,142,78,139,81,136,84,133),(85,154,88,151,91,148,94,145),(86,153,89,150,92,147,95,156),(87,152,90,149,93,146,96,155),(97,157,100,166,103,163,106,160),(98,168,101,165,104,162,107,159),(99,167,102,164,105,161,108,158),(109,172,112,169,115,178,118,175),(110,171,113,180,116,177,119,174),(111,170,114,179,117,176,120,173)], [(1,130,10,127,7,124,4,121),(2,123,11,132,8,129,5,126),(3,128,12,125,9,122,6,131),(13,144,22,141,19,138,16,135),(14,137,23,134,20,143,17,140),(15,142,24,139,21,136,18,133),(25,148,34,145,31,154,28,151),(26,153,35,150,32,147,29,156),(27,146,36,155,33,152,30,149),(37,157,46,166,43,163,40,160),(38,162,47,159,44,168,41,165),(39,167,48,164,45,161,42,158),(49,169,58,178,55,175,52,172),(50,174,59,171,56,180,53,177),(51,179,60,176,57,173,54,170),(61,191,70,188,67,185,64,182),(62,184,71,181,68,190,65,187),(63,189,72,186,69,183,66,192),(73,195,82,204,79,201,76,198),(74,200,83,197,80,194,77,203),(75,193,84,202,81,199,78,196),(85,216,94,213,91,210,88,207),(86,209,95,206,92,215,89,212),(87,214,96,211,93,208,90,205),(97,222,106,219,103,228,100,225),(98,227,107,224,104,221,101,218),(99,220,108,217,105,226,102,223),(109,229,118,238,115,235,112,232),(110,234,119,231,116,240,113,237),(111,239,120,236,117,233,114,230)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | D4 | Dic3 | D6 | C3⋊D4 | C5×S3 | C5×D4 | C5×Dic3 | S3×C10 | C5×C3⋊D4 | C4.10D4 | C12.10D4 | C5×C4.10D4 | C5×C12.10D4 |
| kernel | C5×C12.10D4 | C5×C4.Dic3 | Q8×C30 | C2×C60 | C12.10D4 | C4.Dic3 | C6×Q8 | C2×C12 | Q8×C10 | C60 | C2×C20 | C2×C20 | C20 | C2×Q8 | C12 | C2×C4 | C2×C4 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 8 | 4 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C12.10D4 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 | 0 |
| 0 | 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 0 | 87 |
| 226 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 239 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 84 | 0 | 0 | 0 | 0 |
| 66 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 61 | 141 | 122 |
| 0 | 0 | 80 | 80 | 19 | 160 |
| 0 | 0 | 61 | 80 | 0 | 0 |
| 0 | 0 | 100 | 161 | 161 | 100 |
| 0 | 84 | 0 | 0 | 0 | 0 |
| 175 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 | 240 | 239 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,0,0,0,0,0,0,87,0,0,0,0,0,0,87,0,0,0,0,0,0,87],[226,0,0,0,0,0,0,16,0,0,0,0,0,0,0,240,240,1,0,0,1,0,240,0,0,0,0,0,240,1,0,0,0,0,239,1],[0,66,0,0,0,0,84,0,0,0,0,0,0,0,61,80,61,100,0,0,61,80,80,161,0,0,141,19,0,161,0,0,122,160,0,100],[0,175,0,0,0,0,84,0,0,0,0,0,0,0,0,240,0,1,0,0,0,240,1,0,0,0,1,240,0,0,0,0,0,239,0,1] >;
C5×C12.10D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}._{10}D_4 % in TeX
G:=Group("C5xC12.10D4"); // GroupNames label
G:=SmallGroup(480,155);
// by ID
G=gap.SmallGroup(480,155);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,568,1410,136,4204,15686]);
// 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=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^9*c^3>;
// generators/relations