Copied to
clipboard

## G = Dic3×C2×C10order 240 = 24·3·5

### Direct product of C2×C10 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C10
 Chief series C1 — C3 — C6 — C30 — C5×Dic3 — C10×Dic3 — Dic3×C2×C10
 Lower central C3 — Dic3×C2×C10
 Upper central C1 — C22×C10

Generators and relations for Dic3×C2×C10
G = < a,b,c,d | a2=b10=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152 in 108 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C23, C10, C10 [×6], Dic3 [×4], C2×C6 [×7], C15, C22×C4, C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×C6, C30, C30 [×6], C2×C20 [×6], C22×C10, C22×Dic3, C5×Dic3 [×4], C2×C30 [×7], C22×C20, C10×Dic3 [×6], C22×C30, Dic3×C2×C10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], Dic3 [×4], D6 [×3], C22×C4, C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×S3, C5×S3, C2×C20 [×6], C22×C10, C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C22×C20, C10×Dic3 [×6], S3×C2×C10, Dic3×C2×C10

Smallest permutation representation of Dic3×C2×C10
Regular action on 240 points
Generators in S240
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 192)(12 193)(13 194)(14 195)(15 196)(16 197)(17 198)(18 199)(19 200)(20 191)(21 171)(22 172)(23 173)(24 174)(25 175)(26 176)(27 177)(28 178)(29 179)(30 180)(31 185)(32 186)(33 187)(34 188)(35 189)(36 190)(37 181)(38 182)(39 183)(40 184)(41 228)(42 229)(43 230)(44 221)(45 222)(46 223)(47 224)(48 225)(49 226)(50 227)(51 201)(52 202)(53 203)(54 204)(55 205)(56 206)(57 207)(58 208)(59 209)(60 210)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 161)(68 162)(69 163)(70 164)(71 148)(72 149)(73 150)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 151)(90 152)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 131)(101 127)(102 128)(103 129)(104 130)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(211 240)(212 231)(213 232)(214 233)(215 234)(216 235)(217 236)(218 237)(219 238)(220 239)
(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)
(1 106 95 81 62 76)(2 107 96 82 63 77)(3 108 97 83 64 78)(4 109 98 84 65 79)(5 110 99 85 66 80)(6 101 100 86 67 71)(7 102 91 87 68 72)(8 103 92 88 69 73)(9 104 93 89 70 74)(10 105 94 90 61 75)(11 34 28 59 45 238)(12 35 29 60 46 239)(13 36 30 51 47 240)(14 37 21 52 48 231)(15 38 22 53 49 232)(16 39 23 54 50 233)(17 40 24 55 41 234)(18 31 25 56 42 235)(19 32 26 57 43 236)(20 33 27 58 44 237)(111 121 135 152 165 142)(112 122 136 153 166 143)(113 123 137 154 167 144)(114 124 138 155 168 145)(115 125 139 156 169 146)(116 126 140 157 170 147)(117 127 131 158 161 148)(118 128 132 159 162 149)(119 129 133 160 163 150)(120 130 134 151 164 141)(171 202 225 212 195 181)(172 203 226 213 196 182)(173 204 227 214 197 183)(174 205 228 215 198 184)(175 206 229 216 199 185)(176 207 230 217 200 186)(177 208 221 218 191 187)(178 209 222 219 192 188)(179 210 223 220 193 189)(180 201 224 211 194 190)
(1 218 81 177)(2 219 82 178)(3 220 83 179)(4 211 84 180)(5 212 85 171)(6 213 86 172)(7 214 87 173)(8 215 88 174)(9 216 89 175)(10 217 90 176)(11 137 59 144)(12 138 60 145)(13 139 51 146)(14 140 52 147)(15 131 53 148)(16 132 54 149)(17 133 55 150)(18 134 56 141)(19 135 57 142)(20 136 58 143)(21 116 231 157)(22 117 232 158)(23 118 233 159)(24 119 234 160)(25 120 235 151)(26 111 236 152)(27 112 237 153)(28 113 238 154)(29 114 239 155)(30 115 240 156)(31 130 42 164)(32 121 43 165)(33 122 44 166)(34 123 45 167)(35 124 46 168)(36 125 47 169)(37 126 48 170)(38 127 49 161)(39 128 50 162)(40 129 41 163)(61 186 105 230)(62 187 106 221)(63 188 107 222)(64 189 108 223)(65 190 109 224)(66 181 110 225)(67 182 101 226)(68 183 102 227)(69 184 103 228)(70 185 104 229)(71 196 100 203)(72 197 91 204)(73 198 92 205)(74 199 93 206)(75 200 94 207)(76 191 95 208)(77 192 96 209)(78 193 97 210)(79 194 98 201)(80 195 99 202)

G:=sub<Sym(240)| (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,192)(12,193)(13,194)(14,195)(15,196)(16,197)(17,198)(18,199)(19,200)(20,191)(21,171)(22,172)(23,173)(24,174)(25,175)(26,176)(27,177)(28,178)(29,179)(30,180)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,181)(38,182)(39,183)(40,184)(41,228)(42,229)(43,230)(44,221)(45,222)(46,223)(47,224)(48,225)(49,226)(50,227)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,161)(68,162)(69,163)(70,164)(71,148)(72,149)(73,150)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,151)(90,152)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,131)(101,127)(102,128)(103,129)(104,130)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(211,240)(212,231)(213,232)(214,233)(215,234)(216,235)(217,236)(218,237)(219,238)(220,239), (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), (1,106,95,81,62,76)(2,107,96,82,63,77)(3,108,97,83,64,78)(4,109,98,84,65,79)(5,110,99,85,66,80)(6,101,100,86,67,71)(7,102,91,87,68,72)(8,103,92,88,69,73)(9,104,93,89,70,74)(10,105,94,90,61,75)(11,34,28,59,45,238)(12,35,29,60,46,239)(13,36,30,51,47,240)(14,37,21,52,48,231)(15,38,22,53,49,232)(16,39,23,54,50,233)(17,40,24,55,41,234)(18,31,25,56,42,235)(19,32,26,57,43,236)(20,33,27,58,44,237)(111,121,135,152,165,142)(112,122,136,153,166,143)(113,123,137,154,167,144)(114,124,138,155,168,145)(115,125,139,156,169,146)(116,126,140,157,170,147)(117,127,131,158,161,148)(118,128,132,159,162,149)(119,129,133,160,163,150)(120,130,134,151,164,141)(171,202,225,212,195,181)(172,203,226,213,196,182)(173,204,227,214,197,183)(174,205,228,215,198,184)(175,206,229,216,199,185)(176,207,230,217,200,186)(177,208,221,218,191,187)(178,209,222,219,192,188)(179,210,223,220,193,189)(180,201,224,211,194,190), (1,218,81,177)(2,219,82,178)(3,220,83,179)(4,211,84,180)(5,212,85,171)(6,213,86,172)(7,214,87,173)(8,215,88,174)(9,216,89,175)(10,217,90,176)(11,137,59,144)(12,138,60,145)(13,139,51,146)(14,140,52,147)(15,131,53,148)(16,132,54,149)(17,133,55,150)(18,134,56,141)(19,135,57,142)(20,136,58,143)(21,116,231,157)(22,117,232,158)(23,118,233,159)(24,119,234,160)(25,120,235,151)(26,111,236,152)(27,112,237,153)(28,113,238,154)(29,114,239,155)(30,115,240,156)(31,130,42,164)(32,121,43,165)(33,122,44,166)(34,123,45,167)(35,124,46,168)(36,125,47,169)(37,126,48,170)(38,127,49,161)(39,128,50,162)(40,129,41,163)(61,186,105,230)(62,187,106,221)(63,188,107,222)(64,189,108,223)(65,190,109,224)(66,181,110,225)(67,182,101,226)(68,183,102,227)(69,184,103,228)(70,185,104,229)(71,196,100,203)(72,197,91,204)(73,198,92,205)(74,199,93,206)(75,200,94,207)(76,191,95,208)(77,192,96,209)(78,193,97,210)(79,194,98,201)(80,195,99,202)>;

G:=Group( (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,192)(12,193)(13,194)(14,195)(15,196)(16,197)(17,198)(18,199)(19,200)(20,191)(21,171)(22,172)(23,173)(24,174)(25,175)(26,176)(27,177)(28,178)(29,179)(30,180)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,181)(38,182)(39,183)(40,184)(41,228)(42,229)(43,230)(44,221)(45,222)(46,223)(47,224)(48,225)(49,226)(50,227)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,161)(68,162)(69,163)(70,164)(71,148)(72,149)(73,150)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,151)(90,152)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,131)(101,127)(102,128)(103,129)(104,130)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(211,240)(212,231)(213,232)(214,233)(215,234)(216,235)(217,236)(218,237)(219,238)(220,239), (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), (1,106,95,81,62,76)(2,107,96,82,63,77)(3,108,97,83,64,78)(4,109,98,84,65,79)(5,110,99,85,66,80)(6,101,100,86,67,71)(7,102,91,87,68,72)(8,103,92,88,69,73)(9,104,93,89,70,74)(10,105,94,90,61,75)(11,34,28,59,45,238)(12,35,29,60,46,239)(13,36,30,51,47,240)(14,37,21,52,48,231)(15,38,22,53,49,232)(16,39,23,54,50,233)(17,40,24,55,41,234)(18,31,25,56,42,235)(19,32,26,57,43,236)(20,33,27,58,44,237)(111,121,135,152,165,142)(112,122,136,153,166,143)(113,123,137,154,167,144)(114,124,138,155,168,145)(115,125,139,156,169,146)(116,126,140,157,170,147)(117,127,131,158,161,148)(118,128,132,159,162,149)(119,129,133,160,163,150)(120,130,134,151,164,141)(171,202,225,212,195,181)(172,203,226,213,196,182)(173,204,227,214,197,183)(174,205,228,215,198,184)(175,206,229,216,199,185)(176,207,230,217,200,186)(177,208,221,218,191,187)(178,209,222,219,192,188)(179,210,223,220,193,189)(180,201,224,211,194,190), (1,218,81,177)(2,219,82,178)(3,220,83,179)(4,211,84,180)(5,212,85,171)(6,213,86,172)(7,214,87,173)(8,215,88,174)(9,216,89,175)(10,217,90,176)(11,137,59,144)(12,138,60,145)(13,139,51,146)(14,140,52,147)(15,131,53,148)(16,132,54,149)(17,133,55,150)(18,134,56,141)(19,135,57,142)(20,136,58,143)(21,116,231,157)(22,117,232,158)(23,118,233,159)(24,119,234,160)(25,120,235,151)(26,111,236,152)(27,112,237,153)(28,113,238,154)(29,114,239,155)(30,115,240,156)(31,130,42,164)(32,121,43,165)(33,122,44,166)(34,123,45,167)(35,124,46,168)(36,125,47,169)(37,126,48,170)(38,127,49,161)(39,128,50,162)(40,129,41,163)(61,186,105,230)(62,187,106,221)(63,188,107,222)(64,189,108,223)(65,190,109,224)(66,181,110,225)(67,182,101,226)(68,183,102,227)(69,184,103,228)(70,185,104,229)(71,196,100,203)(72,197,91,204)(73,198,92,205)(74,199,93,206)(75,200,94,207)(76,191,95,208)(77,192,96,209)(78,193,97,210)(79,194,98,201)(80,195,99,202) );

G=PermutationGroup([(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,111),(11,192),(12,193),(13,194),(14,195),(15,196),(16,197),(17,198),(18,199),(19,200),(20,191),(21,171),(22,172),(23,173),(24,174),(25,175),(26,176),(27,177),(28,178),(29,179),(30,180),(31,185),(32,186),(33,187),(34,188),(35,189),(36,190),(37,181),(38,182),(39,183),(40,184),(41,228),(42,229),(43,230),(44,221),(45,222),(46,223),(47,224),(48,225),(49,226),(50,227),(51,201),(52,202),(53,203),(54,204),(55,205),(56,206),(57,207),(58,208),(59,209),(60,210),(61,165),(62,166),(63,167),(64,168),(65,169),(66,170),(67,161),(68,162),(69,163),(70,164),(71,148),(72,149),(73,150),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,151),(90,152),(91,132),(92,133),(93,134),(94,135),(95,136),(96,137),(97,138),(98,139),(99,140),(100,131),(101,127),(102,128),(103,129),(104,130),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(211,240),(212,231),(213,232),(214,233),(215,234),(216,235),(217,236),(218,237),(219,238),(220,239)], [(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)], [(1,106,95,81,62,76),(2,107,96,82,63,77),(3,108,97,83,64,78),(4,109,98,84,65,79),(5,110,99,85,66,80),(6,101,100,86,67,71),(7,102,91,87,68,72),(8,103,92,88,69,73),(9,104,93,89,70,74),(10,105,94,90,61,75),(11,34,28,59,45,238),(12,35,29,60,46,239),(13,36,30,51,47,240),(14,37,21,52,48,231),(15,38,22,53,49,232),(16,39,23,54,50,233),(17,40,24,55,41,234),(18,31,25,56,42,235),(19,32,26,57,43,236),(20,33,27,58,44,237),(111,121,135,152,165,142),(112,122,136,153,166,143),(113,123,137,154,167,144),(114,124,138,155,168,145),(115,125,139,156,169,146),(116,126,140,157,170,147),(117,127,131,158,161,148),(118,128,132,159,162,149),(119,129,133,160,163,150),(120,130,134,151,164,141),(171,202,225,212,195,181),(172,203,226,213,196,182),(173,204,227,214,197,183),(174,205,228,215,198,184),(175,206,229,216,199,185),(176,207,230,217,200,186),(177,208,221,218,191,187),(178,209,222,219,192,188),(179,210,223,220,193,189),(180,201,224,211,194,190)], [(1,218,81,177),(2,219,82,178),(3,220,83,179),(4,211,84,180),(5,212,85,171),(6,213,86,172),(7,214,87,173),(8,215,88,174),(9,216,89,175),(10,217,90,176),(11,137,59,144),(12,138,60,145),(13,139,51,146),(14,140,52,147),(15,131,53,148),(16,132,54,149),(17,133,55,150),(18,134,56,141),(19,135,57,142),(20,136,58,143),(21,116,231,157),(22,117,232,158),(23,118,233,159),(24,119,234,160),(25,120,235,151),(26,111,236,152),(27,112,237,153),(28,113,238,154),(29,114,239,155),(30,115,240,156),(31,130,42,164),(32,121,43,165),(33,122,44,166),(34,123,45,167),(35,124,46,168),(36,125,47,169),(37,126,48,170),(38,127,49,161),(39,128,50,162),(40,129,41,163),(61,186,105,230),(62,187,106,221),(63,188,107,222),(64,189,108,223),(65,190,109,224),(66,181,110,225),(67,182,101,226),(68,183,102,227),(69,184,103,228),(70,185,104,229),(71,196,100,203),(72,197,91,204),(73,198,92,205),(74,199,93,206),(75,200,94,207),(76,191,95,208),(77,192,96,209),(78,193,97,210),(79,194,98,201),(80,195,99,202)])

Dic3×C2×C10 is a maximal subgroup of
C30.24C42  C23.26(S3×D5)  (C2×C30).D4  C10.(C2×D12)  C1528(C4×D4)  (C2×C6)⋊D20  (C2×C10)⋊8Dic6  S3×C22×C20

120 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 15A 15B 15C 15D 20A ··· 20AF 30A ··· 30AB order 1 2 ··· 2 3 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 ··· 1 2 3 ··· 3 1 1 1 1 2 ··· 2 1 ··· 1 2 2 2 2 3 ··· 3 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 S3 Dic3 D6 C5×S3 C5×Dic3 S3×C10 kernel Dic3×C2×C10 C10×Dic3 C22×C30 C2×C30 C22×Dic3 C2×Dic3 C22×C6 C2×C6 C22×C10 C2×C10 C2×C10 C23 C22 C22 # reps 1 6 1 8 4 24 4 32 1 4 3 4 16 12

Matrix representation of Dic3×C2×C10 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 1 60 0 0 1 0
,
 60 0 0 0 0 1 0 0 0 0 57 31 0 0 27 4
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,60,0],[60,0,0,0,0,1,0,0,0,0,57,27,0,0,31,4] >;

Dic3×C2×C10 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{10}
% in TeX

G:=Group("Dic3xC2xC10");
// GroupNames label

G:=SmallGroup(240,173);
// by ID

G=gap.SmallGroup(240,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,240,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽