metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1Dic6, (C6×D5)⋊1Q8, C6.33(Q8×D5), (C2×Dic6)⋊4D5, C30.42(C2×Q8), C3⋊3(D10⋊3Q8), C30.134(C2×D4), C10.130(S3×D4), (C2×C20).226D6, (C2×C12).18D10, C15⋊14(C22⋊Q8), (C5×Dic3).6D4, C2.15(D5×Dic6), (C10×Dic6)⋊14C2, D10⋊C4.8S3, C30.64(C4○D4), C6.Dic10⋊17C2, C30.Q8⋊23C2, C30.4Q8⋊18C2, (C2×Dic5).34D6, C10.15(C2×Dic6), (C22×D5).49D6, (C2×C30).111C23, (C2×C60).319C22, C6.35(Q8⋊2D5), C5⋊6(Dic3.D4), Dic3.8(C5⋊D4), C10.13(D4⋊2S3), C2.17(D20⋊S3), (C2×Dic3).104D10, D10⋊Dic3.11C2, (C6×Dic5).65C22, (C2×Dic15).89C22, (C10×Dic3).69C22, (C2×C4).47(S3×D5), C6.31(C2×C5⋊D4), C2.12(S3×C5⋊D4), (C2×D5×Dic3).5C2, (D5×C2×C6).19C22, C22.177(C2×S3×D5), (C2×C6).123(C22×D5), (C3×D10⋊C4).11C2, (C2×C10).123(C22×S3), SmallGroup(480,497)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊1Dic6
G = < a,b,c,d | a10=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 684 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, D10, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊Q8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C2×C4×D5, Q8×C10, Dic3.D4, D5×Dic3, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D10⋊3Q8, D10⋊Dic3, C30.Q8, C6.Dic10, C3×D10⋊C4, C30.4Q8, C2×D5×Dic3, C10×Dic6, D10⋊1Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C22×S3, C22⋊Q8, C5⋊D4, C22×D5, C2×Dic6, S3×D4, D4⋊2S3, S3×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, Dic3.D4, C2×S3×D5, D10⋊3Q8, D5×Dic6, D20⋊S3, S3×C5⋊D4, D10⋊1Dic6
►On 240 points
Generators in S240
(1 195 100 117 184 135 58 214 129 46)(2 196 101 118 185 136 59 215 130 47)(3 197 102 119 186 137 60 216 131 48)(4 198 103 120 187 138 49 205 132 37)(5 199 104 109 188 139 50 206 121 38)(6 200 105 110 189 140 51 207 122 39)(7 201 106 111 190 141 52 208 123 40)(8 202 107 112 191 142 53 209 124 41)(9 203 108 113 192 143 54 210 125 42)(10 204 97 114 181 144 55 211 126 43)(11 193 98 115 182 133 56 212 127 44)(12 194 99 116 183 134 57 213 128 45)(13 220 153 64 80 163 87 33 239 170)(14 221 154 65 81 164 88 34 240 171)(15 222 155 66 82 165 89 35 229 172)(16 223 156 67 83 166 90 36 230 173)(17 224 145 68 84 167 91 25 231 174)(18 225 146 69 73 168 92 26 232 175)(19 226 147 70 74 157 93 27 233 176)(20 227 148 71 75 158 94 28 234 177)(21 228 149 72 76 159 95 29 235 178)(22 217 150 61 77 160 96 30 236 179)(23 218 151 62 78 161 85 31 237 180)(24 219 152 63 79 162 86 32 238 169)
(1 40)(2 191)(3 42)(4 181)(5 44)(6 183)(7 46)(8 185)(9 48)(10 187)(11 38)(12 189)(13 27)(14 148)(15 29)(16 150)(17 31)(18 152)(19 33)(20 154)(21 35)(22 156)(23 25)(24 146)(26 162)(28 164)(30 166)(32 168)(34 158)(36 160)(37 144)(39 134)(41 136)(43 138)(45 140)(47 142)(49 126)(50 115)(51 128)(52 117)(53 130)(54 119)(55 132)(56 109)(57 122)(58 111)(59 124)(60 113)(61 173)(62 84)(63 175)(64 74)(65 177)(66 76)(67 179)(68 78)(69 169)(70 80)(71 171)(72 82)(73 238)(75 240)(77 230)(79 232)(81 234)(83 236)(85 224)(86 92)(87 226)(88 94)(89 228)(90 96)(91 218)(93 220)(95 222)(97 103)(98 206)(99 105)(100 208)(101 107)(102 210)(104 212)(106 214)(108 216)(110 194)(112 196)(114 198)(116 200)(118 202)(120 204)(121 193)(123 195)(125 197)(127 199)(129 201)(131 203)(133 188)(135 190)(137 192)(139 182)(141 184)(143 186)(145 161)(147 163)(149 165)(151 167)(153 157)(155 159)(170 233)(172 235)(174 237)(176 239)(178 229)(180 231)(205 211)(207 213)(209 215)(217 223)(219 225)(221 227)
(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 7 170)(2 175 8 169)(3 174 9 180)(4 173 10 179)(5 172 11 178)(6 171 12 177)(13 195 19 201)(14 194 20 200)(15 193 21 199)(16 204 22 198)(17 203 23 197)(18 202 24 196)(25 125 31 131)(26 124 32 130)(27 123 33 129)(28 122 34 128)(29 121 35 127)(30 132 36 126)(37 230 43 236)(38 229 44 235)(39 240 45 234)(40 239 46 233)(41 238 47 232)(42 237 48 231)(49 166 55 160)(50 165 56 159)(51 164 57 158)(52 163 58 157)(53 162 59 168)(54 161 60 167)(61 187 67 181)(62 186 68 192)(63 185 69 191)(64 184 70 190)(65 183 71 189)(66 182 72 188)(73 142 79 136)(74 141 80 135)(75 140 81 134)(76 139 82 133)(77 138 83 144)(78 137 84 143)(85 216 91 210)(86 215 92 209)(87 214 93 208)(88 213 94 207)(89 212 95 206)(90 211 96 205)(97 217 103 223)(98 228 104 222)(99 227 105 221)(100 226 106 220)(101 225 107 219)(102 224 108 218)(109 155 115 149)(110 154 116 148)(111 153 117 147)(112 152 118 146)(113 151 119 145)(114 150 120 156)
G:=sub<Sym(240)| (1,195,100,117,184,135,58,214,129,46)(2,196,101,118,185,136,59,215,130,47)(3,197,102,119,186,137,60,216,131,48)(4,198,103,120,187,138,49,205,132,37)(5,199,104,109,188,139,50,206,121,38)(6,200,105,110,189,140,51,207,122,39)(7,201,106,111,190,141,52,208,123,40)(8,202,107,112,191,142,53,209,124,41)(9,203,108,113,192,143,54,210,125,42)(10,204,97,114,181,144,55,211,126,43)(11,193,98,115,182,133,56,212,127,44)(12,194,99,116,183,134,57,213,128,45)(13,220,153,64,80,163,87,33,239,170)(14,221,154,65,81,164,88,34,240,171)(15,222,155,66,82,165,89,35,229,172)(16,223,156,67,83,166,90,36,230,173)(17,224,145,68,84,167,91,25,231,174)(18,225,146,69,73,168,92,26,232,175)(19,226,147,70,74,157,93,27,233,176)(20,227,148,71,75,158,94,28,234,177)(21,228,149,72,76,159,95,29,235,178)(22,217,150,61,77,160,96,30,236,179)(23,218,151,62,78,161,85,31,237,180)(24,219,152,63,79,162,86,32,238,169), (1,40)(2,191)(3,42)(4,181)(5,44)(6,183)(7,46)(8,185)(9,48)(10,187)(11,38)(12,189)(13,27)(14,148)(15,29)(16,150)(17,31)(18,152)(19,33)(20,154)(21,35)(22,156)(23,25)(24,146)(26,162)(28,164)(30,166)(32,168)(34,158)(36,160)(37,144)(39,134)(41,136)(43,138)(45,140)(47,142)(49,126)(50,115)(51,128)(52,117)(53,130)(54,119)(55,132)(56,109)(57,122)(58,111)(59,124)(60,113)(61,173)(62,84)(63,175)(64,74)(65,177)(66,76)(67,179)(68,78)(69,169)(70,80)(71,171)(72,82)(73,238)(75,240)(77,230)(79,232)(81,234)(83,236)(85,224)(86,92)(87,226)(88,94)(89,228)(90,96)(91,218)(93,220)(95,222)(97,103)(98,206)(99,105)(100,208)(101,107)(102,210)(104,212)(106,214)(108,216)(110,194)(112,196)(114,198)(116,200)(118,202)(120,204)(121,193)(123,195)(125,197)(127,199)(129,201)(131,203)(133,188)(135,190)(137,192)(139,182)(141,184)(143,186)(145,161)(147,163)(149,165)(151,167)(153,157)(155,159)(170,233)(172,235)(174,237)(176,239)(178,229)(180,231)(205,211)(207,213)(209,215)(217,223)(219,225)(221,227), (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), (1,176,7,170)(2,175,8,169)(3,174,9,180)(4,173,10,179)(5,172,11,178)(6,171,12,177)(13,195,19,201)(14,194,20,200)(15,193,21,199)(16,204,22,198)(17,203,23,197)(18,202,24,196)(25,125,31,131)(26,124,32,130)(27,123,33,129)(28,122,34,128)(29,121,35,127)(30,132,36,126)(37,230,43,236)(38,229,44,235)(39,240,45,234)(40,239,46,233)(41,238,47,232)(42,237,48,231)(49,166,55,160)(50,165,56,159)(51,164,57,158)(52,163,58,157)(53,162,59,168)(54,161,60,167)(61,187,67,181)(62,186,68,192)(63,185,69,191)(64,184,70,190)(65,183,71,189)(66,182,72,188)(73,142,79,136)(74,141,80,135)(75,140,81,134)(76,139,82,133)(77,138,83,144)(78,137,84,143)(85,216,91,210)(86,215,92,209)(87,214,93,208)(88,213,94,207)(89,212,95,206)(90,211,96,205)(97,217,103,223)(98,228,104,222)(99,227,105,221)(100,226,106,220)(101,225,107,219)(102,224,108,218)(109,155,115,149)(110,154,116,148)(111,153,117,147)(112,152,118,146)(113,151,119,145)(114,150,120,156)>;
G:=Group( (1,195,100,117,184,135,58,214,129,46)(2,196,101,118,185,136,59,215,130,47)(3,197,102,119,186,137,60,216,131,48)(4,198,103,120,187,138,49,205,132,37)(5,199,104,109,188,139,50,206,121,38)(6,200,105,110,189,140,51,207,122,39)(7,201,106,111,190,141,52,208,123,40)(8,202,107,112,191,142,53,209,124,41)(9,203,108,113,192,143,54,210,125,42)(10,204,97,114,181,144,55,211,126,43)(11,193,98,115,182,133,56,212,127,44)(12,194,99,116,183,134,57,213,128,45)(13,220,153,64,80,163,87,33,239,170)(14,221,154,65,81,164,88,34,240,171)(15,222,155,66,82,165,89,35,229,172)(16,223,156,67,83,166,90,36,230,173)(17,224,145,68,84,167,91,25,231,174)(18,225,146,69,73,168,92,26,232,175)(19,226,147,70,74,157,93,27,233,176)(20,227,148,71,75,158,94,28,234,177)(21,228,149,72,76,159,95,29,235,178)(22,217,150,61,77,160,96,30,236,179)(23,218,151,62,78,161,85,31,237,180)(24,219,152,63,79,162,86,32,238,169), (1,40)(2,191)(3,42)(4,181)(5,44)(6,183)(7,46)(8,185)(9,48)(10,187)(11,38)(12,189)(13,27)(14,148)(15,29)(16,150)(17,31)(18,152)(19,33)(20,154)(21,35)(22,156)(23,25)(24,146)(26,162)(28,164)(30,166)(32,168)(34,158)(36,160)(37,144)(39,134)(41,136)(43,138)(45,140)(47,142)(49,126)(50,115)(51,128)(52,117)(53,130)(54,119)(55,132)(56,109)(57,122)(58,111)(59,124)(60,113)(61,173)(62,84)(63,175)(64,74)(65,177)(66,76)(67,179)(68,78)(69,169)(70,80)(71,171)(72,82)(73,238)(75,240)(77,230)(79,232)(81,234)(83,236)(85,224)(86,92)(87,226)(88,94)(89,228)(90,96)(91,218)(93,220)(95,222)(97,103)(98,206)(99,105)(100,208)(101,107)(102,210)(104,212)(106,214)(108,216)(110,194)(112,196)(114,198)(116,200)(118,202)(120,204)(121,193)(123,195)(125,197)(127,199)(129,201)(131,203)(133,188)(135,190)(137,192)(139,182)(141,184)(143,186)(145,161)(147,163)(149,165)(151,167)(153,157)(155,159)(170,233)(172,235)(174,237)(176,239)(178,229)(180,231)(205,211)(207,213)(209,215)(217,223)(219,225)(221,227), (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), (1,176,7,170)(2,175,8,169)(3,174,9,180)(4,173,10,179)(5,172,11,178)(6,171,12,177)(13,195,19,201)(14,194,20,200)(15,193,21,199)(16,204,22,198)(17,203,23,197)(18,202,24,196)(25,125,31,131)(26,124,32,130)(27,123,33,129)(28,122,34,128)(29,121,35,127)(30,132,36,126)(37,230,43,236)(38,229,44,235)(39,240,45,234)(40,239,46,233)(41,238,47,232)(42,237,48,231)(49,166,55,160)(50,165,56,159)(51,164,57,158)(52,163,58,157)(53,162,59,168)(54,161,60,167)(61,187,67,181)(62,186,68,192)(63,185,69,191)(64,184,70,190)(65,183,71,189)(66,182,72,188)(73,142,79,136)(74,141,80,135)(75,140,81,134)(76,139,82,133)(77,138,83,144)(78,137,84,143)(85,216,91,210)(86,215,92,209)(87,214,93,208)(88,213,94,207)(89,212,95,206)(90,211,96,205)(97,217,103,223)(98,228,104,222)(99,227,105,221)(100,226,106,220)(101,225,107,219)(102,224,108,218)(109,155,115,149)(110,154,116,148)(111,153,117,147)(112,152,118,146)(113,151,119,145)(114,150,120,156) );
G=PermutationGroup([[(1,195,100,117,184,135,58,214,129,46),(2,196,101,118,185,136,59,215,130,47),(3,197,102,119,186,137,60,216,131,48),(4,198,103,120,187,138,49,205,132,37),(5,199,104,109,188,139,50,206,121,38),(6,200,105,110,189,140,51,207,122,39),(7,201,106,111,190,141,52,208,123,40),(8,202,107,112,191,142,53,209,124,41),(9,203,108,113,192,143,54,210,125,42),(10,204,97,114,181,144,55,211,126,43),(11,193,98,115,182,133,56,212,127,44),(12,194,99,116,183,134,57,213,128,45),(13,220,153,64,80,163,87,33,239,170),(14,221,154,65,81,164,88,34,240,171),(15,222,155,66,82,165,89,35,229,172),(16,223,156,67,83,166,90,36,230,173),(17,224,145,68,84,167,91,25,231,174),(18,225,146,69,73,168,92,26,232,175),(19,226,147,70,74,157,93,27,233,176),(20,227,148,71,75,158,94,28,234,177),(21,228,149,72,76,159,95,29,235,178),(22,217,150,61,77,160,96,30,236,179),(23,218,151,62,78,161,85,31,237,180),(24,219,152,63,79,162,86,32,238,169)], [(1,40),(2,191),(3,42),(4,181),(5,44),(6,183),(7,46),(8,185),(9,48),(10,187),(11,38),(12,189),(13,27),(14,148),(15,29),(16,150),(17,31),(18,152),(19,33),(20,154),(21,35),(22,156),(23,25),(24,146),(26,162),(28,164),(30,166),(32,168),(34,158),(36,160),(37,144),(39,134),(41,136),(43,138),(45,140),(47,142),(49,126),(50,115),(51,128),(52,117),(53,130),(54,119),(55,132),(56,109),(57,122),(58,111),(59,124),(60,113),(61,173),(62,84),(63,175),(64,74),(65,177),(66,76),(67,179),(68,78),(69,169),(70,80),(71,171),(72,82),(73,238),(75,240),(77,230),(79,232),(81,234),(83,236),(85,224),(86,92),(87,226),(88,94),(89,228),(90,96),(91,218),(93,220),(95,222),(97,103),(98,206),(99,105),(100,208),(101,107),(102,210),(104,212),(106,214),(108,216),(110,194),(112,196),(114,198),(116,200),(118,202),(120,204),(121,193),(123,195),(125,197),(127,199),(129,201),(131,203),(133,188),(135,190),(137,192),(139,182),(141,184),(143,186),(145,161),(147,163),(149,165),(151,167),(153,157),(155,159),(170,233),(172,235),(174,237),(176,239),(178,229),(180,231),(205,211),(207,213),(209,215),(217,223),(219,225),(221,227)], [(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)], [(1,176,7,170),(2,175,8,169),(3,174,9,180),(4,173,10,179),(5,172,11,178),(6,171,12,177),(13,195,19,201),(14,194,20,200),(15,193,21,199),(16,204,22,198),(17,203,23,197),(18,202,24,196),(25,125,31,131),(26,124,32,130),(27,123,33,129),(28,122,34,128),(29,121,35,127),(30,132,36,126),(37,230,43,236),(38,229,44,235),(39,240,45,234),(40,239,46,233),(41,238,47,232),(42,237,48,231),(49,166,55,160),(50,165,56,159),(51,164,57,158),(52,163,58,157),(53,162,59,168),(54,161,60,167),(61,187,67,181),(62,186,68,192),(63,185,69,191),(64,184,70,190),(65,183,71,189),(66,182,72,188),(73,142,79,136),(74,141,80,135),(75,140,81,134),(76,139,82,133),(77,138,83,144),(78,137,84,143),(85,216,91,210),(86,215,92,209),(87,214,93,208),(88,213,94,207),(89,212,95,206),(90,211,96,205),(97,217,103,223),(98,228,104,222),(99,227,105,221),(100,226,106,220),(101,225,107,219),(102,224,108,218),(109,155,115,149),(110,154,116,148),(111,153,117,147),(112,152,118,146),(113,151,119,145),(114,150,120,156)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 4 | 6 | 6 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | D6 | C4○D4 | D10 | D10 | Dic6 | C5⋊D4 | S3×D4 | D4⋊2S3 | S3×D5 | Q8×D5 | Q8⋊2D5 | C2×S3×D5 | D5×Dic6 | D20⋊S3 | S3×C5⋊D4 |
| kernel | D10⋊1Dic6 | D10⋊Dic3 | C30.Q8 | C6.Dic10 | C3×D10⋊C4 | C30.4Q8 | C2×D5×Dic3 | C10×Dic6 | D10⋊C4 | C5×Dic3 | C6×D5 | C2×Dic6 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | D10 | Dic3 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D10⋊1Dic6 ►in GL6(𝔽61)
| 17 | 17 | 0 | 0 | 0 | 0 |
| 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 17 | 17 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 60 |
| 47 | 16 | 0 | 0 | 0 | 0 |
| 45 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 5 |
| 0 | 0 | 0 | 0 | 58 | 40 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 8 | 0 | 0 |
| 0 | 0 | 25 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 54 | 50 |
G:=sub<GL(6,GF(61))| [17,44,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[17,1,0,0,0,0,17,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,60],[47,45,0,0,0,0,16,14,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,21,58,0,0,0,0,5,40],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,25,0,0,0,0,8,17,0,0,0,0,0,0,11,54,0,0,0,0,0,50] >;
D10⋊1Dic6 in GAP, Magma, Sage, TeX
D_{10}\rtimes_1{\rm Dic}_6 % in TeX
G:=Group("D10:1Dic6"); // GroupNames label
G:=SmallGroup(480,497);
// by ID
G=gap.SmallGroup(480,497);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations