direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C10.D4, C30.6Q8, C30.39D4, Dic5⋊1C12, C6.6Dic10, C15⋊9(C4⋊C4), (C2×C20).1C6, (C2×C60).1C2, C2.4(D5×C12), C10.5(C3×D4), C6.18(C4×D5), (C2×C12).1D5, C10.1(C3×Q8), C30.43(C2×C4), (C3×Dic5)⋊3C4, (C2×C6).32D10, C22.4(C6×D5), C10.10(C2×C12), C6.21(C5⋊D4), (C2×Dic5).1C6, (C6×Dic5).6C2, C2.1(C3×Dic10), (C2×C30).33C22, C5⋊2(C3×C4⋊C4), (C2×C4).1(C3×D5), C2.1(C3×C5⋊D4), (C2×C10).4(C2×C6), SmallGroup(240,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C10.D4
G = < a,b,c,d | a3=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C3×C10.D4
►Regular action on 240 points
Generators in S240
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(11 224 204)(12 225 205)(13 226 206)(14 227 207)(15 228 208)(16 229 209)(17 230 210)(18 221 201)(19 222 202)(20 223 203)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 130 101)(82 121 102)(83 122 103)(84 123 104)(85 124 105)(86 125 106)(87 126 107)(88 127 108)(89 128 109)(90 129 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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)
(1 150 25 136)(2 149 26 135)(3 148 27 134)(4 147 28 133)(5 146 29 132)(6 145 30 131)(7 144 21 140)(8 143 22 139)(9 142 23 138)(10 141 24 137)(11 130 240 120)(12 129 231 119)(13 128 232 118)(14 127 233 117)(15 126 234 116)(16 125 235 115)(17 124 236 114)(18 123 237 113)(19 122 238 112)(20 121 239 111)(31 163 42 159)(32 162 43 158)(33 161 44 157)(34 170 45 156)(35 169 46 155)(36 168 47 154)(37 167 48 153)(38 166 49 152)(39 165 50 151)(40 164 41 160)(51 183 62 179)(52 182 63 178)(53 181 64 177)(54 190 65 176)(55 189 66 175)(56 188 67 174)(57 187 68 173)(58 186 69 172)(59 185 70 171)(60 184 61 180)(71 203 82 199)(72 202 83 198)(73 201 84 197)(74 210 85 196)(75 209 86 195)(76 208 87 194)(77 207 88 193)(78 206 89 192)(79 205 90 191)(80 204 81 200)(91 223 102 219)(92 222 103 218)(93 221 104 217)(94 230 105 216)(95 229 106 215)(96 228 107 214)(97 227 108 213)(98 226 109 212)(99 225 110 211)(100 224 101 220)
(1 191 6 196)(2 200 7 195)(3 199 8 194)(4 198 9 193)(5 197 10 192)(11 61 16 66)(12 70 17 65)(13 69 18 64)(14 68 19 63)(15 67 20 62)(21 209 26 204)(22 208 27 203)(23 207 28 202)(24 206 29 201)(25 205 30 210)(31 214 36 219)(32 213 37 218)(33 212 38 217)(34 211 39 216)(35 220 40 215)(41 229 46 224)(42 228 47 223)(43 227 48 222)(44 226 49 221)(45 225 50 230)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 144 86 149)(82 143 87 148)(83 142 88 147)(84 141 89 146)(85 150 90 145)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(101 164 106 169)(102 163 107 168)(103 162 108 167)(104 161 109 166)(105 170 110 165)(111 179 116 174)(112 178 117 173)(113 177 118 172)(114 176 119 171)(115 175 120 180)(121 183 126 188)(122 182 127 187)(123 181 128 186)(124 190 129 185)(125 189 130 184)
G:=sub<Sym(240)| (1,54,34)(2,55,35)(3,56,36)(4,57,37)(5,58,38)(6,59,39)(7,60,40)(8,51,31)(9,52,32)(10,53,33)(11,224,204)(12,225,205)(13,226,206)(14,227,207)(15,228,208)(16,229,209)(17,230,210)(18,221,201)(19,222,202)(20,223,203)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,130,101)(82,121,102)(83,122,103)(84,123,104)(85,124,105)(86,125,106)(87,126,107)(88,127,108)(89,128,109)(90,129,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,150,25,136)(2,149,26,135)(3,148,27,134)(4,147,28,133)(5,146,29,132)(6,145,30,131)(7,144,21,140)(8,143,22,139)(9,142,23,138)(10,141,24,137)(11,130,240,120)(12,129,231,119)(13,128,232,118)(14,127,233,117)(15,126,234,116)(16,125,235,115)(17,124,236,114)(18,123,237,113)(19,122,238,112)(20,121,239,111)(31,163,42,159)(32,162,43,158)(33,161,44,157)(34,170,45,156)(35,169,46,155)(36,168,47,154)(37,167,48,153)(38,166,49,152)(39,165,50,151)(40,164,41,160)(51,183,62,179)(52,182,63,178)(53,181,64,177)(54,190,65,176)(55,189,66,175)(56,188,67,174)(57,187,68,173)(58,186,69,172)(59,185,70,171)(60,184,61,180)(71,203,82,199)(72,202,83,198)(73,201,84,197)(74,210,85,196)(75,209,86,195)(76,208,87,194)(77,207,88,193)(78,206,89,192)(79,205,90,191)(80,204,81,200)(91,223,102,219)(92,222,103,218)(93,221,104,217)(94,230,105,216)(95,229,106,215)(96,228,107,214)(97,227,108,213)(98,226,109,212)(99,225,110,211)(100,224,101,220), (1,191,6,196)(2,200,7,195)(3,199,8,194)(4,198,9,193)(5,197,10,192)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,214,36,219)(32,213,37,218)(33,212,38,217)(34,211,39,216)(35,220,40,215)(41,229,46,224)(42,228,47,223)(43,227,48,222)(44,226,49,221)(45,225,50,230)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(101,164,106,169)(102,163,107,168)(103,162,108,167)(104,161,109,166)(105,170,110,165)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,183,126,188)(122,182,127,187)(123,181,128,186)(124,190,129,185)(125,189,130,184)>;
G:=Group( (1,54,34)(2,55,35)(3,56,36)(4,57,37)(5,58,38)(6,59,39)(7,60,40)(8,51,31)(9,52,32)(10,53,33)(11,224,204)(12,225,205)(13,226,206)(14,227,207)(15,228,208)(16,229,209)(17,230,210)(18,221,201)(19,222,202)(20,223,203)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,130,101)(82,121,102)(83,122,103)(84,123,104)(85,124,105)(86,125,106)(87,126,107)(88,127,108)(89,128,109)(90,129,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,150,25,136)(2,149,26,135)(3,148,27,134)(4,147,28,133)(5,146,29,132)(6,145,30,131)(7,144,21,140)(8,143,22,139)(9,142,23,138)(10,141,24,137)(11,130,240,120)(12,129,231,119)(13,128,232,118)(14,127,233,117)(15,126,234,116)(16,125,235,115)(17,124,236,114)(18,123,237,113)(19,122,238,112)(20,121,239,111)(31,163,42,159)(32,162,43,158)(33,161,44,157)(34,170,45,156)(35,169,46,155)(36,168,47,154)(37,167,48,153)(38,166,49,152)(39,165,50,151)(40,164,41,160)(51,183,62,179)(52,182,63,178)(53,181,64,177)(54,190,65,176)(55,189,66,175)(56,188,67,174)(57,187,68,173)(58,186,69,172)(59,185,70,171)(60,184,61,180)(71,203,82,199)(72,202,83,198)(73,201,84,197)(74,210,85,196)(75,209,86,195)(76,208,87,194)(77,207,88,193)(78,206,89,192)(79,205,90,191)(80,204,81,200)(91,223,102,219)(92,222,103,218)(93,221,104,217)(94,230,105,216)(95,229,106,215)(96,228,107,214)(97,227,108,213)(98,226,109,212)(99,225,110,211)(100,224,101,220), (1,191,6,196)(2,200,7,195)(3,199,8,194)(4,198,9,193)(5,197,10,192)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,209,26,204)(22,208,27,203)(23,207,28,202)(24,206,29,201)(25,205,30,210)(31,214,36,219)(32,213,37,218)(33,212,38,217)(34,211,39,216)(35,220,40,215)(41,229,46,224)(42,228,47,223)(43,227,48,222)(44,226,49,221)(45,225,50,230)(51,234,56,239)(52,233,57,238)(53,232,58,237)(54,231,59,236)(55,240,60,235)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,144,86,149)(82,143,87,148)(83,142,88,147)(84,141,89,146)(85,150,90,145)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(101,164,106,169)(102,163,107,168)(103,162,108,167)(104,161,109,166)(105,170,110,165)(111,179,116,174)(112,178,117,173)(113,177,118,172)(114,176,119,171)(115,175,120,180)(121,183,126,188)(122,182,127,187)(123,181,128,186)(124,190,129,185)(125,189,130,184) );
G=PermutationGroup([[(1,54,34),(2,55,35),(3,56,36),(4,57,37),(5,58,38),(6,59,39),(7,60,40),(8,51,31),(9,52,32),(10,53,33),(11,224,204),(12,225,205),(13,226,206),(14,227,207),(15,228,208),(16,229,209),(17,230,210),(18,221,201),(19,222,202),(20,223,203),(21,61,41),(22,62,42),(23,63,43),(24,64,44),(25,65,45),(26,66,46),(27,67,47),(28,68,48),(29,69,49),(30,70,50),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100),(81,130,101),(82,121,102),(83,122,103),(84,123,104),(85,124,105),(86,125,106),(87,126,107),(88,127,108),(89,128,109),(90,129,110),(131,171,151),(132,172,152),(133,173,153),(134,174,154),(135,175,155),(136,176,156),(137,177,157),(138,178,158),(139,179,159),(140,180,160),(141,181,161),(142,182,162),(143,183,163),(144,184,164),(145,185,165),(146,186,166),(147,187,167),(148,188,168),(149,189,169),(150,190,170),(191,231,211),(192,232,212),(193,233,213),(194,234,214),(195,235,215),(196,236,216),(197,237,217),(198,238,218),(199,239,219),(200,240,220)], [(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)], [(1,150,25,136),(2,149,26,135),(3,148,27,134),(4,147,28,133),(5,146,29,132),(6,145,30,131),(7,144,21,140),(8,143,22,139),(9,142,23,138),(10,141,24,137),(11,130,240,120),(12,129,231,119),(13,128,232,118),(14,127,233,117),(15,126,234,116),(16,125,235,115),(17,124,236,114),(18,123,237,113),(19,122,238,112),(20,121,239,111),(31,163,42,159),(32,162,43,158),(33,161,44,157),(34,170,45,156),(35,169,46,155),(36,168,47,154),(37,167,48,153),(38,166,49,152),(39,165,50,151),(40,164,41,160),(51,183,62,179),(52,182,63,178),(53,181,64,177),(54,190,65,176),(55,189,66,175),(56,188,67,174),(57,187,68,173),(58,186,69,172),(59,185,70,171),(60,184,61,180),(71,203,82,199),(72,202,83,198),(73,201,84,197),(74,210,85,196),(75,209,86,195),(76,208,87,194),(77,207,88,193),(78,206,89,192),(79,205,90,191),(80,204,81,200),(91,223,102,219),(92,222,103,218),(93,221,104,217),(94,230,105,216),(95,229,106,215),(96,228,107,214),(97,227,108,213),(98,226,109,212),(99,225,110,211),(100,224,101,220)], [(1,191,6,196),(2,200,7,195),(3,199,8,194),(4,198,9,193),(5,197,10,192),(11,61,16,66),(12,70,17,65),(13,69,18,64),(14,68,19,63),(15,67,20,62),(21,209,26,204),(22,208,27,203),(23,207,28,202),(24,206,29,201),(25,205,30,210),(31,214,36,219),(32,213,37,218),(33,212,38,217),(34,211,39,216),(35,220,40,215),(41,229,46,224),(42,228,47,223),(43,227,48,222),(44,226,49,221),(45,225,50,230),(51,234,56,239),(52,233,57,238),(53,232,58,237),(54,231,59,236),(55,240,60,235),(71,139,76,134),(72,138,77,133),(73,137,78,132),(74,136,79,131),(75,135,80,140),(81,144,86,149),(82,143,87,148),(83,142,88,147),(84,141,89,146),(85,150,90,145),(91,159,96,154),(92,158,97,153),(93,157,98,152),(94,156,99,151),(95,155,100,160),(101,164,106,169),(102,163,107,168),(103,162,108,167),(104,161,109,166),(105,170,110,165),(111,179,116,174),(112,178,117,173),(113,177,118,172),(114,176,119,171),(115,175,120,180),(121,183,126,188),(122,182,127,187),(123,181,128,186),(124,190,129,185),(125,189,130,184)]])
C3×C10.D4 is a maximal subgroup of
Dic3⋊5Dic10 Dic15⋊5Q8 Dic3⋊Dic10 Dic15⋊Q8 Dic5.1Dic6 Dic5.2Dic6 Dic15.Q8 C4⋊Dic3⋊D5 C60⋊5C4⋊C2 Dic3.Dic10 D6⋊Dic5⋊C2 D6⋊Dic10 D30.35D4 D6⋊Dic5.C2 D30⋊8Q8 Dic3.3Dic10 C10.D4⋊S3 Dic15.4Q8 (S3×Dic5)⋊C4 D30.23(C2×C4) D30.Q8 Dic15⋊14D4 Dic5⋊D12 D6⋊2Dic10 D30⋊2Q8 D30⋊D4 (C2×D12).D5 D30⋊3Q8 D6⋊3Dic10 D30.6D4 C15⋊22(C4×D4) C12×Dic10 C3×D5×C4⋊C4 C12×C5⋊D4
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | D5 | D10 | C3×D4 | C3×Q8 | C3×D5 | Dic10 | C4×D5 | C5⋊D4 | C6×D5 | C3×Dic10 | D5×C12 | C3×C5⋊D4 |
| kernel | C3×C10.D4 | C6×Dic5 | C2×C60 | C10.D4 | C3×Dic5 | C2×Dic5 | C2×C20 | Dic5 | C30 | C30 | C2×C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C10.D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 18 | 0 | 0 |
| 44 | 44 | 0 | 0 |
| 0 | 0 | 1 | 44 |
| 0 | 0 | 17 | 17 |
| 0 | 43 | 0 | 0 |
| 44 | 0 | 0 | 0 |
| 0 | 0 | 53 | 51 |
| 0 | 0 | 37 | 8 |
| 0 | 46 | 0 | 0 |
| 57 | 0 | 0 | 0 |
| 0 | 0 | 58 | 20 |
| 0 | 0 | 30 | 3 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[0,44,0,0,18,44,0,0,0,0,1,17,0,0,44,17],[0,44,0,0,43,0,0,0,0,0,53,37,0,0,51,8],[0,57,0,0,46,0,0,0,0,0,58,30,0,0,20,3] >;
C3×C10.D4 in GAP, Magma, Sage, TeX
C_3\times C_{10}.D_4 % in TeX
G:=Group("C3xC10.D4"); // GroupNames label
G:=SmallGroup(240,41);
// by ID
G=gap.SmallGroup(240,41);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,144,313,79,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^10=c^4=1,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
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