direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×Dic5, C15⋊4C42, Dic15⋊4C4, C5⋊3(C4×Dic3), C3⋊1(C4×Dic5), C6.12(C4×D5), (C2×C6).5D10, (C2×C10).5D6, C10.19(C4×S3), C30.28(C2×C4), (C3×Dic5)⋊1C4, (C5×Dic3)⋊3C4, C2.2(S3×Dic5), C6.3(C2×Dic5), C2.2(D5×Dic3), C22.4(S3×D5), (C2×C30).2C22, (C2×Dic3).5D5, (C6×Dic5).1C2, (C2×Dic5).7S3, C2.2(D30.C2), (C2×Dic15).4C2, (C10×Dic3).1C2, C10.10(C2×Dic3), SmallGroup(240,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — Dic3×Dic5 |
Generators and relations for Dic3×Dic5
G = < a,b,c,d | a6=c10=1, b2=a3, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 176 in 60 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, Dic5, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C30, C2×Dic5, C2×Dic5, C2×C20, C4×Dic3, C5×Dic3, C3×Dic5, Dic15, C2×C30, C4×Dic5, C6×Dic5, C10×Dic3, C2×Dic15, Dic3×Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, Dic5, D10, C4×S3, C2×Dic3, C4×D5, C2×Dic5, C4×Dic3, S3×D5, C4×Dic5, D5×Dic3, S3×Dic5, D30.C2, Dic3×Dic5
►Regular action on 240 points
Generators in S240
(1 68 28 59 47 19)(2 69 29 60 48 20)(3 70 30 51 49 11)(4 61 21 52 50 12)(5 62 22 53 41 13)(6 63 23 54 42 14)(7 64 24 55 43 15)(8 65 25 56 44 16)(9 66 26 57 45 17)(10 67 27 58 46 18)(31 194 213 228 236 202)(32 195 214 229 237 203)(33 196 215 230 238 204)(34 197 216 221 239 205)(35 198 217 222 240 206)(36 199 218 223 231 207)(37 200 219 224 232 208)(38 191 220 225 233 209)(39 192 211 226 234 210)(40 193 212 227 235 201)(71 97 102 114 83 130)(72 98 103 115 84 121)(73 99 104 116 85 122)(74 100 105 117 86 123)(75 91 106 118 87 124)(76 92 107 119 88 125)(77 93 108 120 89 126)(78 94 109 111 90 127)(79 95 110 112 81 128)(80 96 101 113 82 129)(131 189 149 173 165 160)(132 190 150 174 166 151)(133 181 141 175 167 152)(134 182 142 176 168 153)(135 183 143 177 169 154)(136 184 144 178 170 155)(137 185 145 179 161 156)(138 186 146 180 162 157)(139 187 147 171 163 158)(140 188 148 172 164 159)
(1 114 59 71)(2 115 60 72)(3 116 51 73)(4 117 52 74)(5 118 53 75)(6 119 54 76)(7 120 55 77)(8 111 56 78)(9 112 57 79)(10 113 58 80)(11 85 30 99)(12 86 21 100)(13 87 22 91)(14 88 23 92)(15 89 24 93)(16 90 25 94)(17 81 26 95)(18 82 27 96)(19 83 28 97)(20 84 29 98)(31 187 228 163)(32 188 229 164)(33 189 230 165)(34 190 221 166)(35 181 222 167)(36 182 223 168)(37 183 224 169)(38 184 225 170)(39 185 226 161)(40 186 227 162)(41 124 62 106)(42 125 63 107)(43 126 64 108)(44 127 65 109)(45 128 66 110)(46 129 67 101)(47 130 68 102)(48 121 69 103)(49 122 70 104)(50 123 61 105)(131 238 173 196)(132 239 174 197)(133 240 175 198)(134 231 176 199)(135 232 177 200)(136 233 178 191)(137 234 179 192)(138 235 180 193)(139 236 171 194)(140 237 172 195)(141 217 152 206)(142 218 153 207)(143 219 154 208)(144 220 155 209)(145 211 156 210)(146 212 157 201)(147 213 158 202)(148 214 159 203)(149 215 160 204)(150 216 151 205)
(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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 150 16 145)(12 149 17 144)(13 148 18 143)(14 147 19 142)(15 146 20 141)(21 160 26 155)(22 159 27 154)(23 158 28 153)(24 157 29 152)(25 156 30 151)(31 102 36 107)(32 101 37 106)(33 110 38 105)(34 109 39 104)(35 108 40 103)(41 188 46 183)(42 187 47 182)(43 186 48 181)(44 185 49 190)(45 184 50 189)(51 132 56 137)(52 131 57 136)(53 140 58 135)(54 139 59 134)(55 138 60 133)(61 165 66 170)(62 164 67 169)(63 163 68 168)(64 162 69 167)(65 161 70 166)(71 231 76 236)(72 240 77 235)(73 239 78 234)(74 238 79 233)(75 237 80 232)(81 220 86 215)(82 219 87 214)(83 218 88 213)(84 217 89 212)(85 216 90 211)(91 203 96 208)(92 202 97 207)(93 201 98 206)(94 210 99 205)(95 209 100 204)(111 192 116 197)(112 191 117 196)(113 200 118 195)(114 199 119 194)(115 198 120 193)(121 222 126 227)(122 221 127 226)(123 230 128 225)(124 229 129 224)(125 228 130 223)
G:=sub<Sym(240)| (1,68,28,59,47,19)(2,69,29,60,48,20)(3,70,30,51,49,11)(4,61,21,52,50,12)(5,62,22,53,41,13)(6,63,23,54,42,14)(7,64,24,55,43,15)(8,65,25,56,44,16)(9,66,26,57,45,17)(10,67,27,58,46,18)(31,194,213,228,236,202)(32,195,214,229,237,203)(33,196,215,230,238,204)(34,197,216,221,239,205)(35,198,217,222,240,206)(36,199,218,223,231,207)(37,200,219,224,232,208)(38,191,220,225,233,209)(39,192,211,226,234,210)(40,193,212,227,235,201)(71,97,102,114,83,130)(72,98,103,115,84,121)(73,99,104,116,85,122)(74,100,105,117,86,123)(75,91,106,118,87,124)(76,92,107,119,88,125)(77,93,108,120,89,126)(78,94,109,111,90,127)(79,95,110,112,81,128)(80,96,101,113,82,129)(131,189,149,173,165,160)(132,190,150,174,166,151)(133,181,141,175,167,152)(134,182,142,176,168,153)(135,183,143,177,169,154)(136,184,144,178,170,155)(137,185,145,179,161,156)(138,186,146,180,162,157)(139,187,147,171,163,158)(140,188,148,172,164,159), (1,114,59,71)(2,115,60,72)(3,116,51,73)(4,117,52,74)(5,118,53,75)(6,119,54,76)(7,120,55,77)(8,111,56,78)(9,112,57,79)(10,113,58,80)(11,85,30,99)(12,86,21,100)(13,87,22,91)(14,88,23,92)(15,89,24,93)(16,90,25,94)(17,81,26,95)(18,82,27,96)(19,83,28,97)(20,84,29,98)(31,187,228,163)(32,188,229,164)(33,189,230,165)(34,190,221,166)(35,181,222,167)(36,182,223,168)(37,183,224,169)(38,184,225,170)(39,185,226,161)(40,186,227,162)(41,124,62,106)(42,125,63,107)(43,126,64,108)(44,127,65,109)(45,128,66,110)(46,129,67,101)(47,130,68,102)(48,121,69,103)(49,122,70,104)(50,123,61,105)(131,238,173,196)(132,239,174,197)(133,240,175,198)(134,231,176,199)(135,232,177,200)(136,233,178,191)(137,234,179,192)(138,235,180,193)(139,236,171,194)(140,237,172,195)(141,217,152,206)(142,218,153,207)(143,219,154,208)(144,220,155,209)(145,211,156,210)(146,212,157,201)(147,213,158,202)(148,214,159,203)(149,215,160,204)(150,216,151,205), (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), (1,176,6,171)(2,175,7,180)(3,174,8,179)(4,173,9,178)(5,172,10,177)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,160,26,155)(22,159,27,154)(23,158,28,153)(24,157,29,152)(25,156,30,151)(31,102,36,107)(32,101,37,106)(33,110,38,105)(34,109,39,104)(35,108,40,103)(41,188,46,183)(42,187,47,182)(43,186,48,181)(44,185,49,190)(45,184,50,189)(51,132,56,137)(52,131,57,136)(53,140,58,135)(54,139,59,134)(55,138,60,133)(61,165,66,170)(62,164,67,169)(63,163,68,168)(64,162,69,167)(65,161,70,166)(71,231,76,236)(72,240,77,235)(73,239,78,234)(74,238,79,233)(75,237,80,232)(81,220,86,215)(82,219,87,214)(83,218,88,213)(84,217,89,212)(85,216,90,211)(91,203,96,208)(92,202,97,207)(93,201,98,206)(94,210,99,205)(95,209,100,204)(111,192,116,197)(112,191,117,196)(113,200,118,195)(114,199,119,194)(115,198,120,193)(121,222,126,227)(122,221,127,226)(123,230,128,225)(124,229,129,224)(125,228,130,223)>;
G:=Group( (1,68,28,59,47,19)(2,69,29,60,48,20)(3,70,30,51,49,11)(4,61,21,52,50,12)(5,62,22,53,41,13)(6,63,23,54,42,14)(7,64,24,55,43,15)(8,65,25,56,44,16)(9,66,26,57,45,17)(10,67,27,58,46,18)(31,194,213,228,236,202)(32,195,214,229,237,203)(33,196,215,230,238,204)(34,197,216,221,239,205)(35,198,217,222,240,206)(36,199,218,223,231,207)(37,200,219,224,232,208)(38,191,220,225,233,209)(39,192,211,226,234,210)(40,193,212,227,235,201)(71,97,102,114,83,130)(72,98,103,115,84,121)(73,99,104,116,85,122)(74,100,105,117,86,123)(75,91,106,118,87,124)(76,92,107,119,88,125)(77,93,108,120,89,126)(78,94,109,111,90,127)(79,95,110,112,81,128)(80,96,101,113,82,129)(131,189,149,173,165,160)(132,190,150,174,166,151)(133,181,141,175,167,152)(134,182,142,176,168,153)(135,183,143,177,169,154)(136,184,144,178,170,155)(137,185,145,179,161,156)(138,186,146,180,162,157)(139,187,147,171,163,158)(140,188,148,172,164,159), (1,114,59,71)(2,115,60,72)(3,116,51,73)(4,117,52,74)(5,118,53,75)(6,119,54,76)(7,120,55,77)(8,111,56,78)(9,112,57,79)(10,113,58,80)(11,85,30,99)(12,86,21,100)(13,87,22,91)(14,88,23,92)(15,89,24,93)(16,90,25,94)(17,81,26,95)(18,82,27,96)(19,83,28,97)(20,84,29,98)(31,187,228,163)(32,188,229,164)(33,189,230,165)(34,190,221,166)(35,181,222,167)(36,182,223,168)(37,183,224,169)(38,184,225,170)(39,185,226,161)(40,186,227,162)(41,124,62,106)(42,125,63,107)(43,126,64,108)(44,127,65,109)(45,128,66,110)(46,129,67,101)(47,130,68,102)(48,121,69,103)(49,122,70,104)(50,123,61,105)(131,238,173,196)(132,239,174,197)(133,240,175,198)(134,231,176,199)(135,232,177,200)(136,233,178,191)(137,234,179,192)(138,235,180,193)(139,236,171,194)(140,237,172,195)(141,217,152,206)(142,218,153,207)(143,219,154,208)(144,220,155,209)(145,211,156,210)(146,212,157,201)(147,213,158,202)(148,214,159,203)(149,215,160,204)(150,216,151,205), (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), (1,176,6,171)(2,175,7,180)(3,174,8,179)(4,173,9,178)(5,172,10,177)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,160,26,155)(22,159,27,154)(23,158,28,153)(24,157,29,152)(25,156,30,151)(31,102,36,107)(32,101,37,106)(33,110,38,105)(34,109,39,104)(35,108,40,103)(41,188,46,183)(42,187,47,182)(43,186,48,181)(44,185,49,190)(45,184,50,189)(51,132,56,137)(52,131,57,136)(53,140,58,135)(54,139,59,134)(55,138,60,133)(61,165,66,170)(62,164,67,169)(63,163,68,168)(64,162,69,167)(65,161,70,166)(71,231,76,236)(72,240,77,235)(73,239,78,234)(74,238,79,233)(75,237,80,232)(81,220,86,215)(82,219,87,214)(83,218,88,213)(84,217,89,212)(85,216,90,211)(91,203,96,208)(92,202,97,207)(93,201,98,206)(94,210,99,205)(95,209,100,204)(111,192,116,197)(112,191,117,196)(113,200,118,195)(114,199,119,194)(115,198,120,193)(121,222,126,227)(122,221,127,226)(123,230,128,225)(124,229,129,224)(125,228,130,223) );
G=PermutationGroup([[(1,68,28,59,47,19),(2,69,29,60,48,20),(3,70,30,51,49,11),(4,61,21,52,50,12),(5,62,22,53,41,13),(6,63,23,54,42,14),(7,64,24,55,43,15),(8,65,25,56,44,16),(9,66,26,57,45,17),(10,67,27,58,46,18),(31,194,213,228,236,202),(32,195,214,229,237,203),(33,196,215,230,238,204),(34,197,216,221,239,205),(35,198,217,222,240,206),(36,199,218,223,231,207),(37,200,219,224,232,208),(38,191,220,225,233,209),(39,192,211,226,234,210),(40,193,212,227,235,201),(71,97,102,114,83,130),(72,98,103,115,84,121),(73,99,104,116,85,122),(74,100,105,117,86,123),(75,91,106,118,87,124),(76,92,107,119,88,125),(77,93,108,120,89,126),(78,94,109,111,90,127),(79,95,110,112,81,128),(80,96,101,113,82,129),(131,189,149,173,165,160),(132,190,150,174,166,151),(133,181,141,175,167,152),(134,182,142,176,168,153),(135,183,143,177,169,154),(136,184,144,178,170,155),(137,185,145,179,161,156),(138,186,146,180,162,157),(139,187,147,171,163,158),(140,188,148,172,164,159)], [(1,114,59,71),(2,115,60,72),(3,116,51,73),(4,117,52,74),(5,118,53,75),(6,119,54,76),(7,120,55,77),(8,111,56,78),(9,112,57,79),(10,113,58,80),(11,85,30,99),(12,86,21,100),(13,87,22,91),(14,88,23,92),(15,89,24,93),(16,90,25,94),(17,81,26,95),(18,82,27,96),(19,83,28,97),(20,84,29,98),(31,187,228,163),(32,188,229,164),(33,189,230,165),(34,190,221,166),(35,181,222,167),(36,182,223,168),(37,183,224,169),(38,184,225,170),(39,185,226,161),(40,186,227,162),(41,124,62,106),(42,125,63,107),(43,126,64,108),(44,127,65,109),(45,128,66,110),(46,129,67,101),(47,130,68,102),(48,121,69,103),(49,122,70,104),(50,123,61,105),(131,238,173,196),(132,239,174,197),(133,240,175,198),(134,231,176,199),(135,232,177,200),(136,233,178,191),(137,234,179,192),(138,235,180,193),(139,236,171,194),(140,237,172,195),(141,217,152,206),(142,218,153,207),(143,219,154,208),(144,220,155,209),(145,211,156,210),(146,212,157,201),(147,213,158,202),(148,214,159,203),(149,215,160,204),(150,216,151,205)], [(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)], [(1,176,6,171),(2,175,7,180),(3,174,8,179),(4,173,9,178),(5,172,10,177),(11,150,16,145),(12,149,17,144),(13,148,18,143),(14,147,19,142),(15,146,20,141),(21,160,26,155),(22,159,27,154),(23,158,28,153),(24,157,29,152),(25,156,30,151),(31,102,36,107),(32,101,37,106),(33,110,38,105),(34,109,39,104),(35,108,40,103),(41,188,46,183),(42,187,47,182),(43,186,48,181),(44,185,49,190),(45,184,50,189),(51,132,56,137),(52,131,57,136),(53,140,58,135),(54,139,59,134),(55,138,60,133),(61,165,66,170),(62,164,67,169),(63,163,68,168),(64,162,69,167),(65,161,70,166),(71,231,76,236),(72,240,77,235),(73,239,78,234),(74,238,79,233),(75,237,80,232),(81,220,86,215),(82,219,87,214),(83,218,88,213),(84,217,89,212),(85,216,90,211),(91,203,96,208),(92,202,97,207),(93,201,98,206),(94,210,99,205),(95,209,100,204),(111,192,116,197),(112,191,117,196),(113,200,118,195),(114,199,119,194),(115,198,120,193),(121,222,126,227),(122,221,127,226),(123,230,128,225),(124,229,129,224),(125,228,130,223)]])
Dic3×Dic5 is a maximal subgroup of
C30.M4(2) C30.4M4(2) Dic15⋊C8 Dic5⋊5Dic6 Dic3⋊5Dic10 Dic15⋊5Q8 (C2×C20).D6 Dic15⋊1Q8 Dic3⋊Dic10 Dic15⋊Q8 Dic15⋊6Q8 Dic30⋊17C4 Dic5.1Dic6 Dic5.2Dic6 Dic15.Q8 C4⋊Dic3⋊D5 (S3×C20)⋊5C4 Dic15.2Q8 Dic30⋊14C4 D6⋊C4.D5 C60⋊5C4⋊C2 Dic3.Dic10 Dic15⋊7Q8 C4⋊Dic5⋊S3 Dic3.2Dic10 (C4×D15)⋊8C4 (C4×D5)⋊Dic3 (C2×C12).D10 (C2×C60).C22 C4×D5×Dic3 (D5×Dic3)⋊C4 D10.19(C4×S3) Dic15⋊13D4 C4×S3×Dic5 D6.(C4×D5) (S3×Dic5)⋊C4 C4×D30.C2 D30.C2⋊C4 D30.23(C2×C4) Dic15⋊14D4 D6⋊(C4×D5) C15⋊17(C4×D4) C15⋊20(C4×D4) C15⋊22(C4×D4) (C2×Dic6)⋊D5 Dic15.10D4 C23.D5⋊S3 Dic15.19D4 (C6×Dic5)⋊7C4 C23.26(S3×D5) C23.13(S3×D5) C23.14(S3×D5) C23.48(S3×D5) C6.(D4×D5) C15⋊26(C4×D4) C15⋊28(C4×D4) Dic15⋊16D4 Dic15⋊17D4 Dic15⋊5D4
Dic3×Dic5 is a maximal quotient of
Dic15⋊4C8 C30.21C42 C30.22C42 C30.23C42 C30.24C42
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | C4×S3 | C4×D5 | S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 |
| kernel | Dic3×Dic5 | C6×Dic5 | C10×Dic3 | C2×Dic15 | C5×Dic3 | C3×Dic5 | Dic15 | C2×Dic5 | C2×Dic3 | Dic5 | C2×C10 | Dic3 | C2×C6 | C10 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 2 | 2 | 2 | 2 |
Matrix representation of Dic3×Dic5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 15 |
| 0 | 0 | 12 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 30 |
| 0 | 0 | 32 | 25 |
| 0 | 60 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 23 | 54 | 0 | 0 |
| 6 | 38 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,12,0,0,15,2],[1,0,0,0,0,1,0,0,0,0,36,32,0,0,30,25],[0,1,0,0,60,18,0,0,0,0,60,0,0,0,0,60],[23,6,0,0,54,38,0,0,0,0,50,0,0,0,0,50] >;
Dic3×Dic5 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times {\rm Dic}_5 % in TeX
G:=Group("Dic3xDic5"); // GroupNames label
G:=SmallGroup(240,25);
// by ID
G=gap.SmallGroup(240,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^10=1,b^2=a^3,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations