metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊3Dic10, Dic15.8D4, D6⋊C4.4D5, (S3×C10)⋊3Q8, C6.61(D4×D5), C5⋊3(D6⋊Q8), C10.63(S3×D4), (C2×C20).24D6, C30.49(C2×Q8), C10.35(S3×Q8), (C2×Dic30)⋊2C2, C10.D4⋊7S3, C30.138(C2×D4), (C2×C12).24D10, C15⋊20(C22⋊Q8), C30.71(C4○D4), D6⋊Dic5.13C2, (C2×C60).11C22, C6.Dic10⋊23C2, Dic15⋊5C4⋊21C2, (C2×Dic5).40D6, C2.17(S3×Dic10), C6.17(C2×Dic10), C10.58(C4○D12), C6.26(D4⋊2D5), (C2×C30).122C23, (C2×Dic3).40D10, (C22×S3).42D10, C2.14(D10⋊D6), C2.14(D12⋊5D5), C3⋊2(Dic5.14D4), (C6×Dic5).75C22, (C10×Dic3).76C22, (C2×Dic15).99C22, (C5×D6⋊C4).4C2, (C2×C4).55(S3×D5), (C2×S3×Dic5).7C2, C22.185(C2×S3×D5), (C3×C10.D4)⋊7C2, (S3×C2×C10).23C22, (C2×C6).134(C22×D5), (C2×C10).134(C22×S3), SmallGroup(480,508)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊3Dic10
G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=dad-1=a-1, ac=ca, cbc-1=a3b, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 716 in 148 conjugacy classes, 50 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, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D6⋊Q8, S3×Dic5, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, Dic15⋊5C4, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C2×S3×Dic5, C2×Dic30, D6⋊3Dic10
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, C22⋊Q8, Dic10, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, D4⋊2D5, D6⋊Q8, C2×S3×D5, Dic5.14D4, S3×Dic10, D12⋊5D5, D10⋊D6, D6⋊3Dic10
►On 240 points
Generators in S240
(1 62 117 186 229 214)(2 63 118 187 230 215)(3 64 119 188 231 216)(4 65 120 189 232 217)(5 66 101 190 233 218)(6 67 102 191 234 219)(7 68 103 192 235 220)(8 69 104 193 236 201)(9 70 105 194 237 202)(10 71 106 195 238 203)(11 72 107 196 239 204)(12 73 108 197 240 205)(13 74 109 198 221 206)(14 75 110 199 222 207)(15 76 111 200 223 208)(16 77 112 181 224 209)(17 78 113 182 225 210)(18 79 114 183 226 211)(19 80 115 184 227 212)(20 61 116 185 228 213)(21 143 172 99 139 53)(22 144 173 100 140 54)(23 145 174 81 121 55)(24 146 175 82 122 56)(25 147 176 83 123 57)(26 148 177 84 124 58)(27 149 178 85 125 59)(28 150 179 86 126 60)(29 151 180 87 127 41)(30 152 161 88 128 42)(31 153 162 89 129 43)(32 154 163 90 130 44)(33 155 164 91 131 45)(34 156 165 92 132 46)(35 157 166 93 133 47)(36 158 167 94 134 48)(37 159 168 95 135 49)(38 160 169 96 136 50)(39 141 170 97 137 51)(40 142 171 98 138 52)
(1 204)(2 108)(3 206)(4 110)(5 208)(6 112)(7 210)(8 114)(9 212)(10 116)(11 214)(12 118)(13 216)(14 120)(15 218)(16 102)(17 220)(18 104)(19 202)(20 106)(21 31)(22 90)(23 33)(24 92)(25 35)(26 94)(27 37)(28 96)(29 39)(30 98)(32 100)(34 82)(36 84)(38 86)(40 88)(41 141)(42 138)(43 143)(44 140)(45 145)(46 122)(47 147)(48 124)(49 149)(50 126)(51 151)(52 128)(53 153)(54 130)(55 155)(56 132)(57 157)(58 134)(59 159)(60 136)(61 71)(62 239)(63 73)(64 221)(65 75)(66 223)(67 77)(68 225)(69 79)(70 227)(72 229)(74 231)(76 233)(78 235)(80 237)(81 91)(83 93)(85 95)(87 97)(89 99)(101 200)(103 182)(105 184)(107 186)(109 188)(111 190)(113 192)(115 194)(117 196)(119 198)(121 164)(123 166)(125 168)(127 170)(129 172)(131 174)(133 176)(135 178)(137 180)(139 162)(142 161)(144 163)(146 165)(148 167)(150 169)(152 171)(154 173)(156 175)(158 177)(160 179)(181 219)(183 201)(185 203)(187 205)(189 207)(191 209)(193 211)(195 213)(197 215)(199 217)(222 232)(224 234)(226 236)(228 238)(230 240)
(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 54 11 44)(2 53 12 43)(3 52 13 42)(4 51 14 41)(5 50 15 60)(6 49 16 59)(7 48 17 58)(8 47 18 57)(9 46 19 56)(10 45 20 55)(21 205 31 215)(22 204 32 214)(23 203 33 213)(24 202 34 212)(25 201 35 211)(26 220 36 210)(27 219 37 209)(28 218 38 208)(29 217 39 207)(30 216 40 206)(61 121 71 131)(62 140 72 130)(63 139 73 129)(64 138 74 128)(65 137 75 127)(66 136 76 126)(67 135 77 125)(68 134 78 124)(69 133 79 123)(70 132 80 122)(81 106 91 116)(82 105 92 115)(83 104 93 114)(84 103 94 113)(85 102 95 112)(86 101 96 111)(87 120 97 110)(88 119 98 109)(89 118 99 108)(90 117 100 107)(141 222 151 232)(142 221 152 231)(143 240 153 230)(144 239 154 229)(145 238 155 228)(146 237 156 227)(147 236 157 226)(148 235 158 225)(149 234 159 224)(150 233 160 223)(161 188 171 198)(162 187 172 197)(163 186 173 196)(164 185 174 195)(165 184 175 194)(166 183 176 193)(167 182 177 192)(168 181 178 191)(169 200 179 190)(170 199 180 189)
G:=sub<Sym(240)| (1,62,117,186,229,214)(2,63,118,187,230,215)(3,64,119,188,231,216)(4,65,120,189,232,217)(5,66,101,190,233,218)(6,67,102,191,234,219)(7,68,103,192,235,220)(8,69,104,193,236,201)(9,70,105,194,237,202)(10,71,106,195,238,203)(11,72,107,196,239,204)(12,73,108,197,240,205)(13,74,109,198,221,206)(14,75,110,199,222,207)(15,76,111,200,223,208)(16,77,112,181,224,209)(17,78,113,182,225,210)(18,79,114,183,226,211)(19,80,115,184,227,212)(20,61,116,185,228,213)(21,143,172,99,139,53)(22,144,173,100,140,54)(23,145,174,81,121,55)(24,146,175,82,122,56)(25,147,176,83,123,57)(26,148,177,84,124,58)(27,149,178,85,125,59)(28,150,179,86,126,60)(29,151,180,87,127,41)(30,152,161,88,128,42)(31,153,162,89,129,43)(32,154,163,90,130,44)(33,155,164,91,131,45)(34,156,165,92,132,46)(35,157,166,93,133,47)(36,158,167,94,134,48)(37,159,168,95,135,49)(38,160,169,96,136,50)(39,141,170,97,137,51)(40,142,171,98,138,52), (1,204)(2,108)(3,206)(4,110)(5,208)(6,112)(7,210)(8,114)(9,212)(10,116)(11,214)(12,118)(13,216)(14,120)(15,218)(16,102)(17,220)(18,104)(19,202)(20,106)(21,31)(22,90)(23,33)(24,92)(25,35)(26,94)(27,37)(28,96)(29,39)(30,98)(32,100)(34,82)(36,84)(38,86)(40,88)(41,141)(42,138)(43,143)(44,140)(45,145)(46,122)(47,147)(48,124)(49,149)(50,126)(51,151)(52,128)(53,153)(54,130)(55,155)(56,132)(57,157)(58,134)(59,159)(60,136)(61,71)(62,239)(63,73)(64,221)(65,75)(66,223)(67,77)(68,225)(69,79)(70,227)(72,229)(74,231)(76,233)(78,235)(80,237)(81,91)(83,93)(85,95)(87,97)(89,99)(101,200)(103,182)(105,184)(107,186)(109,188)(111,190)(113,192)(115,194)(117,196)(119,198)(121,164)(123,166)(125,168)(127,170)(129,172)(131,174)(133,176)(135,178)(137,180)(139,162)(142,161)(144,163)(146,165)(148,167)(150,169)(152,171)(154,173)(156,175)(158,177)(160,179)(181,219)(183,201)(185,203)(187,205)(189,207)(191,209)(193,211)(195,213)(197,215)(199,217)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,54,11,44)(2,53,12,43)(3,52,13,42)(4,51,14,41)(5,50,15,60)(6,49,16,59)(7,48,17,58)(8,47,18,57)(9,46,19,56)(10,45,20,55)(21,205,31,215)(22,204,32,214)(23,203,33,213)(24,202,34,212)(25,201,35,211)(26,220,36,210)(27,219,37,209)(28,218,38,208)(29,217,39,207)(30,216,40,206)(61,121,71,131)(62,140,72,130)(63,139,73,129)(64,138,74,128)(65,137,75,127)(66,136,76,126)(67,135,77,125)(68,134,78,124)(69,133,79,123)(70,132,80,122)(81,106,91,116)(82,105,92,115)(83,104,93,114)(84,103,94,113)(85,102,95,112)(86,101,96,111)(87,120,97,110)(88,119,98,109)(89,118,99,108)(90,117,100,107)(141,222,151,232)(142,221,152,231)(143,240,153,230)(144,239,154,229)(145,238,155,228)(146,237,156,227)(147,236,157,226)(148,235,158,225)(149,234,159,224)(150,233,160,223)(161,188,171,198)(162,187,172,197)(163,186,173,196)(164,185,174,195)(165,184,175,194)(166,183,176,193)(167,182,177,192)(168,181,178,191)(169,200,179,190)(170,199,180,189)>;
G:=Group( (1,62,117,186,229,214)(2,63,118,187,230,215)(3,64,119,188,231,216)(4,65,120,189,232,217)(5,66,101,190,233,218)(6,67,102,191,234,219)(7,68,103,192,235,220)(8,69,104,193,236,201)(9,70,105,194,237,202)(10,71,106,195,238,203)(11,72,107,196,239,204)(12,73,108,197,240,205)(13,74,109,198,221,206)(14,75,110,199,222,207)(15,76,111,200,223,208)(16,77,112,181,224,209)(17,78,113,182,225,210)(18,79,114,183,226,211)(19,80,115,184,227,212)(20,61,116,185,228,213)(21,143,172,99,139,53)(22,144,173,100,140,54)(23,145,174,81,121,55)(24,146,175,82,122,56)(25,147,176,83,123,57)(26,148,177,84,124,58)(27,149,178,85,125,59)(28,150,179,86,126,60)(29,151,180,87,127,41)(30,152,161,88,128,42)(31,153,162,89,129,43)(32,154,163,90,130,44)(33,155,164,91,131,45)(34,156,165,92,132,46)(35,157,166,93,133,47)(36,158,167,94,134,48)(37,159,168,95,135,49)(38,160,169,96,136,50)(39,141,170,97,137,51)(40,142,171,98,138,52), (1,204)(2,108)(3,206)(4,110)(5,208)(6,112)(7,210)(8,114)(9,212)(10,116)(11,214)(12,118)(13,216)(14,120)(15,218)(16,102)(17,220)(18,104)(19,202)(20,106)(21,31)(22,90)(23,33)(24,92)(25,35)(26,94)(27,37)(28,96)(29,39)(30,98)(32,100)(34,82)(36,84)(38,86)(40,88)(41,141)(42,138)(43,143)(44,140)(45,145)(46,122)(47,147)(48,124)(49,149)(50,126)(51,151)(52,128)(53,153)(54,130)(55,155)(56,132)(57,157)(58,134)(59,159)(60,136)(61,71)(62,239)(63,73)(64,221)(65,75)(66,223)(67,77)(68,225)(69,79)(70,227)(72,229)(74,231)(76,233)(78,235)(80,237)(81,91)(83,93)(85,95)(87,97)(89,99)(101,200)(103,182)(105,184)(107,186)(109,188)(111,190)(113,192)(115,194)(117,196)(119,198)(121,164)(123,166)(125,168)(127,170)(129,172)(131,174)(133,176)(135,178)(137,180)(139,162)(142,161)(144,163)(146,165)(148,167)(150,169)(152,171)(154,173)(156,175)(158,177)(160,179)(181,219)(183,201)(185,203)(187,205)(189,207)(191,209)(193,211)(195,213)(197,215)(199,217)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,54,11,44)(2,53,12,43)(3,52,13,42)(4,51,14,41)(5,50,15,60)(6,49,16,59)(7,48,17,58)(8,47,18,57)(9,46,19,56)(10,45,20,55)(21,205,31,215)(22,204,32,214)(23,203,33,213)(24,202,34,212)(25,201,35,211)(26,220,36,210)(27,219,37,209)(28,218,38,208)(29,217,39,207)(30,216,40,206)(61,121,71,131)(62,140,72,130)(63,139,73,129)(64,138,74,128)(65,137,75,127)(66,136,76,126)(67,135,77,125)(68,134,78,124)(69,133,79,123)(70,132,80,122)(81,106,91,116)(82,105,92,115)(83,104,93,114)(84,103,94,113)(85,102,95,112)(86,101,96,111)(87,120,97,110)(88,119,98,109)(89,118,99,108)(90,117,100,107)(141,222,151,232)(142,221,152,231)(143,240,153,230)(144,239,154,229)(145,238,155,228)(146,237,156,227)(147,236,157,226)(148,235,158,225)(149,234,159,224)(150,233,160,223)(161,188,171,198)(162,187,172,197)(163,186,173,196)(164,185,174,195)(165,184,175,194)(166,183,176,193)(167,182,177,192)(168,181,178,191)(169,200,179,190)(170,199,180,189) );
G=PermutationGroup([[(1,62,117,186,229,214),(2,63,118,187,230,215),(3,64,119,188,231,216),(4,65,120,189,232,217),(5,66,101,190,233,218),(6,67,102,191,234,219),(7,68,103,192,235,220),(8,69,104,193,236,201),(9,70,105,194,237,202),(10,71,106,195,238,203),(11,72,107,196,239,204),(12,73,108,197,240,205),(13,74,109,198,221,206),(14,75,110,199,222,207),(15,76,111,200,223,208),(16,77,112,181,224,209),(17,78,113,182,225,210),(18,79,114,183,226,211),(19,80,115,184,227,212),(20,61,116,185,228,213),(21,143,172,99,139,53),(22,144,173,100,140,54),(23,145,174,81,121,55),(24,146,175,82,122,56),(25,147,176,83,123,57),(26,148,177,84,124,58),(27,149,178,85,125,59),(28,150,179,86,126,60),(29,151,180,87,127,41),(30,152,161,88,128,42),(31,153,162,89,129,43),(32,154,163,90,130,44),(33,155,164,91,131,45),(34,156,165,92,132,46),(35,157,166,93,133,47),(36,158,167,94,134,48),(37,159,168,95,135,49),(38,160,169,96,136,50),(39,141,170,97,137,51),(40,142,171,98,138,52)], [(1,204),(2,108),(3,206),(4,110),(5,208),(6,112),(7,210),(8,114),(9,212),(10,116),(11,214),(12,118),(13,216),(14,120),(15,218),(16,102),(17,220),(18,104),(19,202),(20,106),(21,31),(22,90),(23,33),(24,92),(25,35),(26,94),(27,37),(28,96),(29,39),(30,98),(32,100),(34,82),(36,84),(38,86),(40,88),(41,141),(42,138),(43,143),(44,140),(45,145),(46,122),(47,147),(48,124),(49,149),(50,126),(51,151),(52,128),(53,153),(54,130),(55,155),(56,132),(57,157),(58,134),(59,159),(60,136),(61,71),(62,239),(63,73),(64,221),(65,75),(66,223),(67,77),(68,225),(69,79),(70,227),(72,229),(74,231),(76,233),(78,235),(80,237),(81,91),(83,93),(85,95),(87,97),(89,99),(101,200),(103,182),(105,184),(107,186),(109,188),(111,190),(113,192),(115,194),(117,196),(119,198),(121,164),(123,166),(125,168),(127,170),(129,172),(131,174),(133,176),(135,178),(137,180),(139,162),(142,161),(144,163),(146,165),(148,167),(150,169),(152,171),(154,173),(156,175),(158,177),(160,179),(181,219),(183,201),(185,203),(187,205),(189,207),(191,209),(193,211),(195,213),(197,215),(199,217),(222,232),(224,234),(226,236),(228,238),(230,240)], [(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)], [(1,54,11,44),(2,53,12,43),(3,52,13,42),(4,51,14,41),(5,50,15,60),(6,49,16,59),(7,48,17,58),(8,47,18,57),(9,46,19,56),(10,45,20,55),(21,205,31,215),(22,204,32,214),(23,203,33,213),(24,202,34,212),(25,201,35,211),(26,220,36,210),(27,219,37,209),(28,218,38,208),(29,217,39,207),(30,216,40,206),(61,121,71,131),(62,140,72,130),(63,139,73,129),(64,138,74,128),(65,137,75,127),(66,136,76,126),(67,135,77,125),(68,134,78,124),(69,133,79,123),(70,132,80,122),(81,106,91,116),(82,105,92,115),(83,104,93,114),(84,103,94,113),(85,102,95,112),(86,101,96,111),(87,120,97,110),(88,119,98,109),(89,118,99,108),(90,117,100,107),(141,222,151,232),(142,221,152,231),(143,240,153,230),(144,239,154,229),(145,238,155,228),(146,237,156,227),(147,236,157,226),(148,235,158,225),(149,234,159,224),(150,233,160,223),(161,188,171,198),(162,187,172,197),(163,186,173,196),(164,185,174,195),(165,184,175,194),(166,183,176,193),(167,182,177,192),(168,181,178,191),(169,200,179,190),(170,199,180,189)]])
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 | Dic10 | C4○D12 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | S3×Dic10 | D12⋊5D5 | D10⋊D6 |
| kernel | D6⋊3Dic10 | D6⋊Dic5 | Dic15⋊5C4 | C6.Dic10 | C3×C10.D4 | C5×D6⋊C4 | C2×S3×Dic5 | C2×Dic30 | C10.D4 | Dic15 | S3×C10 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | 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 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊3Dic10 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 1 |
| 33 | 0 | 0 | 0 |
| 38 | 37 | 0 | 0 |
| 0 | 0 | 38 | 15 |
| 0 | 0 | 46 | 23 |
| 2 | 58 | 0 | 0 |
| 22 | 59 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 50 | 50 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,60,1],[33,38,0,0,0,37,0,0,0,0,38,46,0,0,15,23],[2,22,0,0,58,59,0,0,0,0,11,50,0,0,0,50] >;
D6⋊3Dic10 in GAP, Magma, Sage, TeX
D_6\rtimes_3{\rm Dic}_{10} % in TeX
G:=Group("D6:3Dic10"); // GroupNames label
G:=SmallGroup(480,508);
// by ID
G=gap.SmallGroup(480,508);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,590,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations