metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1Dic10, D6⋊C4.8D5, (S3×C10)⋊1Q8, C5⋊3(D6⋊3Q8), (C2×C20).15D6, C6.131(D4×D5), C30.37(C2×Q8), C10.32(S3×Q8), (C2×Dic10)⋊5S3, C30.131(C2×D4), C15⋊10(C22⋊Q8), (C3×Dic5).7D4, (C6×Dic10)⋊15C2, C30.57(C4○D4), (C2×C12).224D10, D6⋊Dic5.11C2, C6.Dic10⋊16C2, C30.Q8⋊16C2, C30.4Q8⋊15C2, C6.14(C2×Dic10), C2.15(S3×Dic10), C6.12(D4⋊2D5), (C2×C30).100C23, (C2×C60).255C22, (C2×Dic5).106D6, (C2×Dic3).30D10, (C22×S3).36D10, C2.15(D12⋊D5), C10.32(Q8⋊3S3), Dic5.20(C3⋊D4), C3⋊6(Dic5.14D4), (C6×Dic5).57C22, (C2×Dic15).80C22, (C10×Dic3).60C22, (C5×D6⋊C4).8C2, (C2×C4).39(S3×D5), C2.15(D5×C3⋊D4), (C2×S3×Dic5).4C2, C10.33(C2×C3⋊D4), C22.169(C2×S3×D5), (S3×C2×C10).15C22, (C2×C6).112(C22×D5), (C2×C10).112(C22×S3), SmallGroup(480,486)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊1Dic10
G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 668 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D6⋊3Q8, S3×Dic5, C3×Dic10, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, C30.Q8, C6.Dic10, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, C6×Dic10, D6⋊1Dic10
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C3⋊D4, C22×S3, C22⋊Q8, Dic10, C22×D5, S3×Q8, Q8⋊3S3, C2×C3⋊D4, S3×D5, C2×Dic10, D4×D5, D4⋊2D5, D6⋊3Q8, C2×S3×D5, Dic5.14D4, S3×Dic10, D12⋊D5, D5×C3⋊D4, D6⋊1Dic10
►On 240 points
Generators in S240
(1 202 223 183 48 38)(2 203 224 184 49 39)(3 204 225 185 50 40)(4 205 226 186 51 21)(5 206 227 187 52 22)(6 207 228 188 53 23)(7 208 229 189 54 24)(8 209 230 190 55 25)(9 210 231 191 56 26)(10 211 232 192 57 27)(11 212 233 193 58 28)(12 213 234 194 59 29)(13 214 235 195 60 30)(14 215 236 196 41 31)(15 216 237 197 42 32)(16 217 238 198 43 33)(17 218 239 199 44 34)(18 219 240 200 45 35)(19 220 221 181 46 36)(20 201 222 182 47 37)(61 87 145 118 164 122)(62 88 146 119 165 123)(63 89 147 120 166 124)(64 90 148 101 167 125)(65 91 149 102 168 126)(66 92 150 103 169 127)(67 93 151 104 170 128)(68 94 152 105 171 129)(69 95 153 106 172 130)(70 96 154 107 173 131)(71 97 155 108 174 132)(72 98 156 109 175 133)(73 99 157 110 176 134)(74 100 158 111 177 135)(75 81 159 112 178 136)(76 82 160 113 179 137)(77 83 141 114 180 138)(78 84 142 115 161 139)(79 85 143 116 162 140)(80 86 144 117 163 121)
(1 38)(2 224)(3 40)(4 226)(5 22)(6 228)(7 24)(8 230)(9 26)(10 232)(11 28)(12 234)(13 30)(14 236)(15 32)(16 238)(17 34)(18 240)(19 36)(20 222)(21 186)(23 188)(25 190)(27 192)(29 194)(31 196)(33 198)(35 200)(37 182)(39 184)(42 216)(44 218)(46 220)(48 202)(50 204)(52 206)(54 208)(56 210)(58 212)(60 214)(61 118)(63 120)(65 102)(67 104)(69 106)(71 108)(73 110)(75 112)(77 114)(79 116)(81 159)(82 137)(83 141)(84 139)(85 143)(86 121)(87 145)(88 123)(89 147)(90 125)(91 149)(92 127)(93 151)(94 129)(95 153)(96 131)(97 155)(98 133)(99 157)(100 135)(122 164)(124 166)(126 168)(128 170)(130 172)(132 174)(134 176)(136 178)(138 180)(140 162)(142 161)(144 163)(146 165)(148 167)(150 169)(152 171)(154 173)(156 175)(158 177)(160 179)(181 221)(183 223)(185 225)(187 227)(189 229)(191 231)(193 233)(195 235)(197 237)(199 239)
(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)
(1 122 11 132)(2 121 12 131)(3 140 13 130)(4 139 14 129)(5 138 15 128)(6 137 16 127)(7 136 17 126)(8 135 18 125)(9 134 19 124)(10 133 20 123)(21 161 31 171)(22 180 32 170)(23 179 33 169)(24 178 34 168)(25 177 35 167)(26 176 36 166)(27 175 37 165)(28 174 38 164)(29 173 39 163)(30 172 40 162)(41 105 51 115)(42 104 52 114)(43 103 53 113)(44 102 54 112)(45 101 55 111)(46 120 56 110)(47 119 57 109)(48 118 58 108)(49 117 59 107)(50 116 60 106)(61 212 71 202)(62 211 72 201)(63 210 73 220)(64 209 74 219)(65 208 75 218)(66 207 76 217)(67 206 77 216)(68 205 78 215)(69 204 79 214)(70 203 80 213)(81 239 91 229)(82 238 92 228)(83 237 93 227)(84 236 94 226)(85 235 95 225)(86 234 96 224)(87 233 97 223)(88 232 98 222)(89 231 99 221)(90 230 100 240)(141 197 151 187)(142 196 152 186)(143 195 153 185)(144 194 154 184)(145 193 155 183)(146 192 156 182)(147 191 157 181)(148 190 158 200)(149 189 159 199)(150 188 160 198)
G:=sub<Sym(240)| (1,202,223,183,48,38)(2,203,224,184,49,39)(3,204,225,185,50,40)(4,205,226,186,51,21)(5,206,227,187,52,22)(6,207,228,188,53,23)(7,208,229,189,54,24)(8,209,230,190,55,25)(9,210,231,191,56,26)(10,211,232,192,57,27)(11,212,233,193,58,28)(12,213,234,194,59,29)(13,214,235,195,60,30)(14,215,236,196,41,31)(15,216,237,197,42,32)(16,217,238,198,43,33)(17,218,239,199,44,34)(18,219,240,200,45,35)(19,220,221,181,46,36)(20,201,222,182,47,37)(61,87,145,118,164,122)(62,88,146,119,165,123)(63,89,147,120,166,124)(64,90,148,101,167,125)(65,91,149,102,168,126)(66,92,150,103,169,127)(67,93,151,104,170,128)(68,94,152,105,171,129)(69,95,153,106,172,130)(70,96,154,107,173,131)(71,97,155,108,174,132)(72,98,156,109,175,133)(73,99,157,110,176,134)(74,100,158,111,177,135)(75,81,159,112,178,136)(76,82,160,113,179,137)(77,83,141,114,180,138)(78,84,142,115,161,139)(79,85,143,116,162,140)(80,86,144,117,163,121), (1,38)(2,224)(3,40)(4,226)(5,22)(6,228)(7,24)(8,230)(9,26)(10,232)(11,28)(12,234)(13,30)(14,236)(15,32)(16,238)(17,34)(18,240)(19,36)(20,222)(21,186)(23,188)(25,190)(27,192)(29,194)(31,196)(33,198)(35,200)(37,182)(39,184)(42,216)(44,218)(46,220)(48,202)(50,204)(52,206)(54,208)(56,210)(58,212)(60,214)(61,118)(63,120)(65,102)(67,104)(69,106)(71,108)(73,110)(75,112)(77,114)(79,116)(81,159)(82,137)(83,141)(84,139)(85,143)(86,121)(87,145)(88,123)(89,147)(90,125)(91,149)(92,127)(93,151)(94,129)(95,153)(96,131)(97,155)(98,133)(99,157)(100,135)(122,164)(124,166)(126,168)(128,170)(130,172)(132,174)(134,176)(136,178)(138,180)(140,162)(142,161)(144,163)(146,165)(148,167)(150,169)(152,171)(154,173)(156,175)(158,177)(160,179)(181,221)(183,223)(185,225)(187,227)(189,229)(191,231)(193,233)(195,235)(197,237)(199,239), (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), (1,122,11,132)(2,121,12,131)(3,140,13,130)(4,139,14,129)(5,138,15,128)(6,137,16,127)(7,136,17,126)(8,135,18,125)(9,134,19,124)(10,133,20,123)(21,161,31,171)(22,180,32,170)(23,179,33,169)(24,178,34,168)(25,177,35,167)(26,176,36,166)(27,175,37,165)(28,174,38,164)(29,173,39,163)(30,172,40,162)(41,105,51,115)(42,104,52,114)(43,103,53,113)(44,102,54,112)(45,101,55,111)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,212,71,202)(62,211,72,201)(63,210,73,220)(64,209,74,219)(65,208,75,218)(66,207,76,217)(67,206,77,216)(68,205,78,215)(69,204,79,214)(70,203,80,213)(81,239,91,229)(82,238,92,228)(83,237,93,227)(84,236,94,226)(85,235,95,225)(86,234,96,224)(87,233,97,223)(88,232,98,222)(89,231,99,221)(90,230,100,240)(141,197,151,187)(142,196,152,186)(143,195,153,185)(144,194,154,184)(145,193,155,183)(146,192,156,182)(147,191,157,181)(148,190,158,200)(149,189,159,199)(150,188,160,198)>;
G:=Group( (1,202,223,183,48,38)(2,203,224,184,49,39)(3,204,225,185,50,40)(4,205,226,186,51,21)(5,206,227,187,52,22)(6,207,228,188,53,23)(7,208,229,189,54,24)(8,209,230,190,55,25)(9,210,231,191,56,26)(10,211,232,192,57,27)(11,212,233,193,58,28)(12,213,234,194,59,29)(13,214,235,195,60,30)(14,215,236,196,41,31)(15,216,237,197,42,32)(16,217,238,198,43,33)(17,218,239,199,44,34)(18,219,240,200,45,35)(19,220,221,181,46,36)(20,201,222,182,47,37)(61,87,145,118,164,122)(62,88,146,119,165,123)(63,89,147,120,166,124)(64,90,148,101,167,125)(65,91,149,102,168,126)(66,92,150,103,169,127)(67,93,151,104,170,128)(68,94,152,105,171,129)(69,95,153,106,172,130)(70,96,154,107,173,131)(71,97,155,108,174,132)(72,98,156,109,175,133)(73,99,157,110,176,134)(74,100,158,111,177,135)(75,81,159,112,178,136)(76,82,160,113,179,137)(77,83,141,114,180,138)(78,84,142,115,161,139)(79,85,143,116,162,140)(80,86,144,117,163,121), (1,38)(2,224)(3,40)(4,226)(5,22)(6,228)(7,24)(8,230)(9,26)(10,232)(11,28)(12,234)(13,30)(14,236)(15,32)(16,238)(17,34)(18,240)(19,36)(20,222)(21,186)(23,188)(25,190)(27,192)(29,194)(31,196)(33,198)(35,200)(37,182)(39,184)(42,216)(44,218)(46,220)(48,202)(50,204)(52,206)(54,208)(56,210)(58,212)(60,214)(61,118)(63,120)(65,102)(67,104)(69,106)(71,108)(73,110)(75,112)(77,114)(79,116)(81,159)(82,137)(83,141)(84,139)(85,143)(86,121)(87,145)(88,123)(89,147)(90,125)(91,149)(92,127)(93,151)(94,129)(95,153)(96,131)(97,155)(98,133)(99,157)(100,135)(122,164)(124,166)(126,168)(128,170)(130,172)(132,174)(134,176)(136,178)(138,180)(140,162)(142,161)(144,163)(146,165)(148,167)(150,169)(152,171)(154,173)(156,175)(158,177)(160,179)(181,221)(183,223)(185,225)(187,227)(189,229)(191,231)(193,233)(195,235)(197,237)(199,239), (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), (1,122,11,132)(2,121,12,131)(3,140,13,130)(4,139,14,129)(5,138,15,128)(6,137,16,127)(7,136,17,126)(8,135,18,125)(9,134,19,124)(10,133,20,123)(21,161,31,171)(22,180,32,170)(23,179,33,169)(24,178,34,168)(25,177,35,167)(26,176,36,166)(27,175,37,165)(28,174,38,164)(29,173,39,163)(30,172,40,162)(41,105,51,115)(42,104,52,114)(43,103,53,113)(44,102,54,112)(45,101,55,111)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,212,71,202)(62,211,72,201)(63,210,73,220)(64,209,74,219)(65,208,75,218)(66,207,76,217)(67,206,77,216)(68,205,78,215)(69,204,79,214)(70,203,80,213)(81,239,91,229)(82,238,92,228)(83,237,93,227)(84,236,94,226)(85,235,95,225)(86,234,96,224)(87,233,97,223)(88,232,98,222)(89,231,99,221)(90,230,100,240)(141,197,151,187)(142,196,152,186)(143,195,153,185)(144,194,154,184)(145,193,155,183)(146,192,156,182)(147,191,157,181)(148,190,158,200)(149,189,159,199)(150,188,160,198) );
G=PermutationGroup([[(1,202,223,183,48,38),(2,203,224,184,49,39),(3,204,225,185,50,40),(4,205,226,186,51,21),(5,206,227,187,52,22),(6,207,228,188,53,23),(7,208,229,189,54,24),(8,209,230,190,55,25),(9,210,231,191,56,26),(10,211,232,192,57,27),(11,212,233,193,58,28),(12,213,234,194,59,29),(13,214,235,195,60,30),(14,215,236,196,41,31),(15,216,237,197,42,32),(16,217,238,198,43,33),(17,218,239,199,44,34),(18,219,240,200,45,35),(19,220,221,181,46,36),(20,201,222,182,47,37),(61,87,145,118,164,122),(62,88,146,119,165,123),(63,89,147,120,166,124),(64,90,148,101,167,125),(65,91,149,102,168,126),(66,92,150,103,169,127),(67,93,151,104,170,128),(68,94,152,105,171,129),(69,95,153,106,172,130),(70,96,154,107,173,131),(71,97,155,108,174,132),(72,98,156,109,175,133),(73,99,157,110,176,134),(74,100,158,111,177,135),(75,81,159,112,178,136),(76,82,160,113,179,137),(77,83,141,114,180,138),(78,84,142,115,161,139),(79,85,143,116,162,140),(80,86,144,117,163,121)], [(1,38),(2,224),(3,40),(4,226),(5,22),(6,228),(7,24),(8,230),(9,26),(10,232),(11,28),(12,234),(13,30),(14,236),(15,32),(16,238),(17,34),(18,240),(19,36),(20,222),(21,186),(23,188),(25,190),(27,192),(29,194),(31,196),(33,198),(35,200),(37,182),(39,184),(42,216),(44,218),(46,220),(48,202),(50,204),(52,206),(54,208),(56,210),(58,212),(60,214),(61,118),(63,120),(65,102),(67,104),(69,106),(71,108),(73,110),(75,112),(77,114),(79,116),(81,159),(82,137),(83,141),(84,139),(85,143),(86,121),(87,145),(88,123),(89,147),(90,125),(91,149),(92,127),(93,151),(94,129),(95,153),(96,131),(97,155),(98,133),(99,157),(100,135),(122,164),(124,166),(126,168),(128,170),(130,172),(132,174),(134,176),(136,178),(138,180),(140,162),(142,161),(144,163),(146,165),(148,167),(150,169),(152,171),(154,173),(156,175),(158,177),(160,179),(181,221),(183,223),(185,225),(187,227),(189,229),(191,231),(193,233),(195,235),(197,237),(199,239)], [(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)], [(1,122,11,132),(2,121,12,131),(3,140,13,130),(4,139,14,129),(5,138,15,128),(6,137,16,127),(7,136,17,126),(8,135,18,125),(9,134,19,124),(10,133,20,123),(21,161,31,171),(22,180,32,170),(23,179,33,169),(24,178,34,168),(25,177,35,167),(26,176,36,166),(27,175,37,165),(28,174,38,164),(29,173,39,163),(30,172,40,162),(41,105,51,115),(42,104,52,114),(43,103,53,113),(44,102,54,112),(45,101,55,111),(46,120,56,110),(47,119,57,109),(48,118,58,108),(49,117,59,107),(50,116,60,106),(61,212,71,202),(62,211,72,201),(63,210,73,220),(64,209,74,219),(65,208,75,218),(66,207,76,217),(67,206,77,216),(68,205,78,215),(69,204,79,214),(70,203,80,213),(81,239,91,229),(82,238,92,228),(83,237,93,227),(84,236,94,226),(85,235,95,225),(86,234,96,224),(87,233,97,223),(88,232,98,222),(89,231,99,221),(90,230,100,240),(141,197,151,187),(142,196,152,186),(143,195,153,185),(144,194,154,184),(145,193,155,183),(146,192,156,182),(147,191,157,181),(148,190,158,200),(149,189,159,199),(150,188,160,198)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 4 | 10 | 10 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | Dic10 | S3×Q8 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | S3×Dic10 | D12⋊D5 | D5×C3⋊D4 |
| kernel | D6⋊1Dic10 | D6⋊Dic5 | C30.Q8 | C6.Dic10 | C5×D6⋊C4 | C30.4Q8 | C2×S3×Dic5 | C6×Dic10 | C2×Dic10 | C3×Dic5 | S3×C10 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | D6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊1Dic10 ►in GL4(𝔽61) generated by
| 1 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 18 | 0 | 0 |
| 43 | 52 | 0 | 0 |
| 0 | 0 | 54 | 29 |
| 0 | 0 | 32 | 59 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 29 |
| 0 | 0 | 31 | 25 |
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[9,43,0,0,18,52,0,0,0,0,54,32,0,0,29,59],[1,0,0,0,0,1,0,0,0,0,36,31,0,0,29,25] >;
D6⋊1Dic10 in GAP, Magma, Sage, TeX
D_6\rtimes_1{\rm Dic}_{10} % in TeX
G:=Group("D6:1Dic10"); // GroupNames label
G:=SmallGroup(480,486);
// by ID
G=gap.SmallGroup(480,486);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations