metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊4Dic10, Dic15.9D4, D6⋊C4.9D5, C4⋊Dic5⋊7S3, (S3×C10)⋊4Q8, C6.42(D4×D5), C5⋊4(D6⋊Q8), (C2×C20).26D6, C10.43(S3×D4), C30.54(C2×D4), C10.36(S3×Q8), C30.50(C2×Q8), C15⋊21(C22⋊Q8), (C2×C12).228D10, D6⋊Dic5.14C2, C30.4Q8⋊16C2, Dic15⋊5C4⋊22C2, (C2×Dic5).42D6, C2.18(S3×Dic10), C6.18(C2×Dic10), C30.123(C4○D4), C10.72(C4○D12), C6.47(D4⋊2D5), C2.19(C20⋊D6), (C2×C30).126C23, (C2×C60).259C22, (C2×Dic3).41D10, (C22×S3).44D10, C3⋊3(Dic5.14D4), (C6×Dic5).78C22, C2.18(Dic3.D10), (C10×Dic3).79C22, (C2×Dic15).100C22, (C2×C15⋊Q8)⋊11C2, (C5×D6⋊C4).9C2, (C2×C4).57(S3×D5), (C2×S3×Dic5).8C2, (C3×C4⋊Dic5)⋊18C2, C22.187(C2×S3×D5), (S3×C2×C10).25C22, (C2×C6).138(C22×D5), (C2×C10).138(C22×S3), SmallGroup(480,512)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊4Dic10
G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=cac-1=dad-1=a-1, cbc-1=ab, 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, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D6⋊Q8, S3×Dic5, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, Dic15⋊5C4, C3×C4⋊Dic5, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, C2×C15⋊Q8, D6⋊4Dic10
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, C20⋊D6, Dic3.D10, D6⋊4Dic10
►On 240 points
Generators in S240
(1 110 156 185 128 219)(2 220 129 186 157 111)(3 112 158 187 130 201)(4 202 131 188 159 113)(5 114 160 189 132 203)(6 204 133 190 141 115)(7 116 142 191 134 205)(8 206 135 192 143 117)(9 118 144 193 136 207)(10 208 137 194 145 119)(11 120 146 195 138 209)(12 210 139 196 147 101)(13 102 148 197 140 211)(14 212 121 198 149 103)(15 104 150 199 122 213)(16 214 123 200 151 105)(17 106 152 181 124 215)(18 216 125 182 153 107)(19 108 154 183 126 217)(20 218 127 184 155 109)(21 179 66 221 88 50)(22 51 89 222 67 180)(23 161 68 223 90 52)(24 53 91 224 69 162)(25 163 70 225 92 54)(26 55 93 226 71 164)(27 165 72 227 94 56)(28 57 95 228 73 166)(29 167 74 229 96 58)(30 59 97 230 75 168)(31 169 76 231 98 60)(32 41 99 232 77 170)(33 171 78 233 100 42)(34 43 81 234 79 172)(35 173 80 235 82 44)(36 45 83 236 61 174)(37 175 62 237 84 46)(38 47 85 238 63 176)(39 177 64 239 86 48)(40 49 87 240 65 178)
(1 209)(2 139)(3 211)(4 121)(5 213)(6 123)(7 215)(8 125)(9 217)(10 127)(11 219)(12 129)(13 201)(14 131)(15 203)(16 133)(17 205)(18 135)(19 207)(20 137)(21 31)(22 41)(23 33)(24 43)(25 35)(26 45)(27 37)(28 47)(29 39)(30 49)(32 51)(34 53)(36 55)(38 57)(40 59)(42 161)(44 163)(46 165)(48 167)(50 169)(52 171)(54 173)(56 175)(58 177)(60 179)(61 226)(62 94)(63 228)(64 96)(65 230)(66 98)(67 232)(68 100)(69 234)(70 82)(71 236)(72 84)(73 238)(74 86)(75 240)(76 88)(77 222)(78 90)(79 224)(80 92)(81 162)(83 164)(85 166)(87 168)(89 170)(91 172)(93 174)(95 176)(97 178)(99 180)(101 186)(102 130)(103 188)(104 132)(105 190)(106 134)(107 192)(108 136)(109 194)(110 138)(111 196)(112 140)(113 198)(114 122)(115 200)(116 124)(117 182)(118 126)(119 184)(120 128)(141 151)(142 181)(143 153)(144 183)(145 155)(146 185)(147 157)(148 187)(149 159)(150 189)(152 191)(154 193)(156 195)(158 197)(160 199)(202 212)(204 214)(206 216)(208 218)(210 220)(221 231)(223 233)(225 235)(227 237)(229 239)
(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 61 11 71)(2 80 12 70)(3 79 13 69)(4 78 14 68)(5 77 15 67)(6 76 16 66)(7 75 17 65)(8 74 18 64)(9 73 19 63)(10 72 20 62)(21 133 31 123)(22 132 32 122)(23 131 33 121)(24 130 34 140)(25 129 35 139)(26 128 36 138)(27 127 37 137)(28 126 38 136)(29 125 39 135)(30 124 40 134)(41 199 51 189)(42 198 52 188)(43 197 53 187)(44 196 54 186)(45 195 55 185)(46 194 56 184)(47 193 57 183)(48 192 58 182)(49 191 59 181)(50 190 60 200)(81 148 91 158)(82 147 92 157)(83 146 93 156)(84 145 94 155)(85 144 95 154)(86 143 96 153)(87 142 97 152)(88 141 98 151)(89 160 99 150)(90 159 100 149)(101 225 111 235)(102 224 112 234)(103 223 113 233)(104 222 114 232)(105 221 115 231)(106 240 116 230)(107 239 117 229)(108 238 118 228)(109 237 119 227)(110 236 120 226)(161 202 171 212)(162 201 172 211)(163 220 173 210)(164 219 174 209)(165 218 175 208)(166 217 176 207)(167 216 177 206)(168 215 178 205)(169 214 179 204)(170 213 180 203)
G:=sub<Sym(240)| (1,110,156,185,128,219)(2,220,129,186,157,111)(3,112,158,187,130,201)(4,202,131,188,159,113)(5,114,160,189,132,203)(6,204,133,190,141,115)(7,116,142,191,134,205)(8,206,135,192,143,117)(9,118,144,193,136,207)(10,208,137,194,145,119)(11,120,146,195,138,209)(12,210,139,196,147,101)(13,102,148,197,140,211)(14,212,121,198,149,103)(15,104,150,199,122,213)(16,214,123,200,151,105)(17,106,152,181,124,215)(18,216,125,182,153,107)(19,108,154,183,126,217)(20,218,127,184,155,109)(21,179,66,221,88,50)(22,51,89,222,67,180)(23,161,68,223,90,52)(24,53,91,224,69,162)(25,163,70,225,92,54)(26,55,93,226,71,164)(27,165,72,227,94,56)(28,57,95,228,73,166)(29,167,74,229,96,58)(30,59,97,230,75,168)(31,169,76,231,98,60)(32,41,99,232,77,170)(33,171,78,233,100,42)(34,43,81,234,79,172)(35,173,80,235,82,44)(36,45,83,236,61,174)(37,175,62,237,84,46)(38,47,85,238,63,176)(39,177,64,239,86,48)(40,49,87,240,65,178), (1,209)(2,139)(3,211)(4,121)(5,213)(6,123)(7,215)(8,125)(9,217)(10,127)(11,219)(12,129)(13,201)(14,131)(15,203)(16,133)(17,205)(18,135)(19,207)(20,137)(21,31)(22,41)(23,33)(24,43)(25,35)(26,45)(27,37)(28,47)(29,39)(30,49)(32,51)(34,53)(36,55)(38,57)(40,59)(42,161)(44,163)(46,165)(48,167)(50,169)(52,171)(54,173)(56,175)(58,177)(60,179)(61,226)(62,94)(63,228)(64,96)(65,230)(66,98)(67,232)(68,100)(69,234)(70,82)(71,236)(72,84)(73,238)(74,86)(75,240)(76,88)(77,222)(78,90)(79,224)(80,92)(81,162)(83,164)(85,166)(87,168)(89,170)(91,172)(93,174)(95,176)(97,178)(99,180)(101,186)(102,130)(103,188)(104,132)(105,190)(106,134)(107,192)(108,136)(109,194)(110,138)(111,196)(112,140)(113,198)(114,122)(115,200)(116,124)(117,182)(118,126)(119,184)(120,128)(141,151)(142,181)(143,153)(144,183)(145,155)(146,185)(147,157)(148,187)(149,159)(150,189)(152,191)(154,193)(156,195)(158,197)(160,199)(202,212)(204,214)(206,216)(208,218)(210,220)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,61,11,71)(2,80,12,70)(3,79,13,69)(4,78,14,68)(5,77,15,67)(6,76,16,66)(7,75,17,65)(8,74,18,64)(9,73,19,63)(10,72,20,62)(21,133,31,123)(22,132,32,122)(23,131,33,121)(24,130,34,140)(25,129,35,139)(26,128,36,138)(27,127,37,137)(28,126,38,136)(29,125,39,135)(30,124,40,134)(41,199,51,189)(42,198,52,188)(43,197,53,187)(44,196,54,186)(45,195,55,185)(46,194,56,184)(47,193,57,183)(48,192,58,182)(49,191,59,181)(50,190,60,200)(81,148,91,158)(82,147,92,157)(83,146,93,156)(84,145,94,155)(85,144,95,154)(86,143,96,153)(87,142,97,152)(88,141,98,151)(89,160,99,150)(90,159,100,149)(101,225,111,235)(102,224,112,234)(103,223,113,233)(104,222,114,232)(105,221,115,231)(106,240,116,230)(107,239,117,229)(108,238,118,228)(109,237,119,227)(110,236,120,226)(161,202,171,212)(162,201,172,211)(163,220,173,210)(164,219,174,209)(165,218,175,208)(166,217,176,207)(167,216,177,206)(168,215,178,205)(169,214,179,204)(170,213,180,203)>;
G:=Group( (1,110,156,185,128,219)(2,220,129,186,157,111)(3,112,158,187,130,201)(4,202,131,188,159,113)(5,114,160,189,132,203)(6,204,133,190,141,115)(7,116,142,191,134,205)(8,206,135,192,143,117)(9,118,144,193,136,207)(10,208,137,194,145,119)(11,120,146,195,138,209)(12,210,139,196,147,101)(13,102,148,197,140,211)(14,212,121,198,149,103)(15,104,150,199,122,213)(16,214,123,200,151,105)(17,106,152,181,124,215)(18,216,125,182,153,107)(19,108,154,183,126,217)(20,218,127,184,155,109)(21,179,66,221,88,50)(22,51,89,222,67,180)(23,161,68,223,90,52)(24,53,91,224,69,162)(25,163,70,225,92,54)(26,55,93,226,71,164)(27,165,72,227,94,56)(28,57,95,228,73,166)(29,167,74,229,96,58)(30,59,97,230,75,168)(31,169,76,231,98,60)(32,41,99,232,77,170)(33,171,78,233,100,42)(34,43,81,234,79,172)(35,173,80,235,82,44)(36,45,83,236,61,174)(37,175,62,237,84,46)(38,47,85,238,63,176)(39,177,64,239,86,48)(40,49,87,240,65,178), (1,209)(2,139)(3,211)(4,121)(5,213)(6,123)(7,215)(8,125)(9,217)(10,127)(11,219)(12,129)(13,201)(14,131)(15,203)(16,133)(17,205)(18,135)(19,207)(20,137)(21,31)(22,41)(23,33)(24,43)(25,35)(26,45)(27,37)(28,47)(29,39)(30,49)(32,51)(34,53)(36,55)(38,57)(40,59)(42,161)(44,163)(46,165)(48,167)(50,169)(52,171)(54,173)(56,175)(58,177)(60,179)(61,226)(62,94)(63,228)(64,96)(65,230)(66,98)(67,232)(68,100)(69,234)(70,82)(71,236)(72,84)(73,238)(74,86)(75,240)(76,88)(77,222)(78,90)(79,224)(80,92)(81,162)(83,164)(85,166)(87,168)(89,170)(91,172)(93,174)(95,176)(97,178)(99,180)(101,186)(102,130)(103,188)(104,132)(105,190)(106,134)(107,192)(108,136)(109,194)(110,138)(111,196)(112,140)(113,198)(114,122)(115,200)(116,124)(117,182)(118,126)(119,184)(120,128)(141,151)(142,181)(143,153)(144,183)(145,155)(146,185)(147,157)(148,187)(149,159)(150,189)(152,191)(154,193)(156,195)(158,197)(160,199)(202,212)(204,214)(206,216)(208,218)(210,220)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,61,11,71)(2,80,12,70)(3,79,13,69)(4,78,14,68)(5,77,15,67)(6,76,16,66)(7,75,17,65)(8,74,18,64)(9,73,19,63)(10,72,20,62)(21,133,31,123)(22,132,32,122)(23,131,33,121)(24,130,34,140)(25,129,35,139)(26,128,36,138)(27,127,37,137)(28,126,38,136)(29,125,39,135)(30,124,40,134)(41,199,51,189)(42,198,52,188)(43,197,53,187)(44,196,54,186)(45,195,55,185)(46,194,56,184)(47,193,57,183)(48,192,58,182)(49,191,59,181)(50,190,60,200)(81,148,91,158)(82,147,92,157)(83,146,93,156)(84,145,94,155)(85,144,95,154)(86,143,96,153)(87,142,97,152)(88,141,98,151)(89,160,99,150)(90,159,100,149)(101,225,111,235)(102,224,112,234)(103,223,113,233)(104,222,114,232)(105,221,115,231)(106,240,116,230)(107,239,117,229)(108,238,118,228)(109,237,119,227)(110,236,120,226)(161,202,171,212)(162,201,172,211)(163,220,173,210)(164,219,174,209)(165,218,175,208)(166,217,176,207)(167,216,177,206)(168,215,178,205)(169,214,179,204)(170,213,180,203) );
G=PermutationGroup([[(1,110,156,185,128,219),(2,220,129,186,157,111),(3,112,158,187,130,201),(4,202,131,188,159,113),(5,114,160,189,132,203),(6,204,133,190,141,115),(7,116,142,191,134,205),(8,206,135,192,143,117),(9,118,144,193,136,207),(10,208,137,194,145,119),(11,120,146,195,138,209),(12,210,139,196,147,101),(13,102,148,197,140,211),(14,212,121,198,149,103),(15,104,150,199,122,213),(16,214,123,200,151,105),(17,106,152,181,124,215),(18,216,125,182,153,107),(19,108,154,183,126,217),(20,218,127,184,155,109),(21,179,66,221,88,50),(22,51,89,222,67,180),(23,161,68,223,90,52),(24,53,91,224,69,162),(25,163,70,225,92,54),(26,55,93,226,71,164),(27,165,72,227,94,56),(28,57,95,228,73,166),(29,167,74,229,96,58),(30,59,97,230,75,168),(31,169,76,231,98,60),(32,41,99,232,77,170),(33,171,78,233,100,42),(34,43,81,234,79,172),(35,173,80,235,82,44),(36,45,83,236,61,174),(37,175,62,237,84,46),(38,47,85,238,63,176),(39,177,64,239,86,48),(40,49,87,240,65,178)], [(1,209),(2,139),(3,211),(4,121),(5,213),(6,123),(7,215),(8,125),(9,217),(10,127),(11,219),(12,129),(13,201),(14,131),(15,203),(16,133),(17,205),(18,135),(19,207),(20,137),(21,31),(22,41),(23,33),(24,43),(25,35),(26,45),(27,37),(28,47),(29,39),(30,49),(32,51),(34,53),(36,55),(38,57),(40,59),(42,161),(44,163),(46,165),(48,167),(50,169),(52,171),(54,173),(56,175),(58,177),(60,179),(61,226),(62,94),(63,228),(64,96),(65,230),(66,98),(67,232),(68,100),(69,234),(70,82),(71,236),(72,84),(73,238),(74,86),(75,240),(76,88),(77,222),(78,90),(79,224),(80,92),(81,162),(83,164),(85,166),(87,168),(89,170),(91,172),(93,174),(95,176),(97,178),(99,180),(101,186),(102,130),(103,188),(104,132),(105,190),(106,134),(107,192),(108,136),(109,194),(110,138),(111,196),(112,140),(113,198),(114,122),(115,200),(116,124),(117,182),(118,126),(119,184),(120,128),(141,151),(142,181),(143,153),(144,183),(145,155),(146,185),(147,157),(148,187),(149,159),(150,189),(152,191),(154,193),(156,195),(158,197),(160,199),(202,212),(204,214),(206,216),(208,218),(210,220),(221,231),(223,233),(225,235),(227,237),(229,239)], [(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)], [(1,61,11,71),(2,80,12,70),(3,79,13,69),(4,78,14,68),(5,77,15,67),(6,76,16,66),(7,75,17,65),(8,74,18,64),(9,73,19,63),(10,72,20,62),(21,133,31,123),(22,132,32,122),(23,131,33,121),(24,130,34,140),(25,129,35,139),(26,128,36,138),(27,127,37,137),(28,126,38,136),(29,125,39,135),(30,124,40,134),(41,199,51,189),(42,198,52,188),(43,197,53,187),(44,196,54,186),(45,195,55,185),(46,194,56,184),(47,193,57,183),(48,192,58,182),(49,191,59,181),(50,190,60,200),(81,148,91,158),(82,147,92,157),(83,146,93,156),(84,145,94,155),(85,144,95,154),(86,143,96,153),(87,142,97,152),(88,141,98,151),(89,160,99,150),(90,159,100,149),(101,225,111,235),(102,224,112,234),(103,223,113,233),(104,222,114,232),(105,221,115,231),(106,240,116,230),(107,239,117,229),(108,238,118,228),(109,237,119,227),(110,236,120,226),(161,202,171,212),(162,201,172,211),(163,220,173,210),(164,219,174,209),(165,218,175,208),(166,217,176,207),(167,216,177,206),(168,215,178,205),(169,214,179,204),(170,213,180,203)]])
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 | C20⋊D6 | Dic3.D10 |
| kernel | D6⋊4Dic10 | D6⋊Dic5 | Dic15⋊5C4 | C3×C4⋊Dic5 | C5×D6⋊C4 | C30.4Q8 | C2×S3×Dic5 | C2×C15⋊Q8 | C4⋊Dic5 | 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⋊4Dic10 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 41 |
| 0 | 0 | 0 | 0 | 52 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 41 |
| 0 | 0 | 0 | 0 | 52 | 2 |
| 54 | 29 | 0 | 0 | 0 | 0 |
| 32 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 30 | 0 | 0 |
| 0 | 0 | 26 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 60 |
| 56 | 51 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 28 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 52 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,52,0,0,0,0,41,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,0,0,59,52,0,0,0,0,41,2],[54,32,0,0,0,0,29,59,0,0,0,0,0,0,16,26,0,0,0,0,30,45,0,0,0,0,0,0,1,9,0,0,0,0,0,60],[56,27,0,0,0,0,51,5,0,0,0,0,0,0,50,28,0,0,0,0,0,11,0,0,0,0,0,0,60,52,0,0,0,0,0,1] >;
D6⋊4Dic10 in GAP, Magma, Sage, TeX
D_6\rtimes_4{\rm Dic}_{10} % in TeX
G:=Group("D6:4Dic10"); // GroupNames label
G:=SmallGroup(480,512);
// by ID
G=gap.SmallGroup(480,512);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,590,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations