metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊2D12, (S3×C10)⋊1D4, (C2×D12)⋊2D5, C6.17(D4×D5), C5⋊4(C12⋊D4), D6⋊1(C5⋊D4), C15⋊3(C4⋊D4), D6⋊Dic5⋊9C2, (C3×Dic5)⋊1D4, C30.41(C2×D4), C2.19(D5×D12), (C10×D12)⋊14C2, C10.129(S3×D4), (C2×C20).224D6, (C2×C12).14D10, C10.17(C2×D12), D30⋊3C4⋊20C2, C30.61(C4○D4), C10.D4⋊16S3, C3⋊1(Dic5⋊D4), C6.13(D4⋊2D5), (C2×C60).317C22, (C2×C30).106C23, (C2×Dic5).107D6, (C22×S3).38D10, C2.16(D12⋊D5), C10.33(Q8⋊3S3), (C6×Dic5).61C22, (C2×Dic15).86C22, (C22×D15).34C22, (C2×S3×Dic5)⋊4C2, (C2×C5⋊D12)⋊1C2, (C2×C4).43(S3×D5), C6.30(C2×C5⋊D4), C2.11(S3×C5⋊D4), C22.173(C2×S3×D5), (S3×C2×C10).18C22, (C3×C10.D4)⋊19C2, (C2×C6).118(C22×D5), (C2×C10).118(C22×S3), SmallGroup(480,492)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊D12
G = < a,b,c,d | a10=c12=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c-1 >
Subgroups: 1100 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, C12⋊D4, S3×Dic5, C5⋊D12, C6×Dic5, C5×D12, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic5⋊D4, D6⋊Dic5, C3×C10.D4, D30⋊3C4, C2×S3×Dic5, C2×C5⋊D12, C10×D12, Dic5⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C2×D12, S3×D4, Q8⋊3S3, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C12⋊D4, C2×S3×D5, Dic5⋊D4, D12⋊D5, D5×D12, S3×C5⋊D4, Dic5⋊D12
►On 240 points
Generators in S240
(1 97 59 75 155 124 233 169 42 210)(2 98 60 76 156 125 234 170 43 211)(3 99 49 77 145 126 235 171 44 212)(4 100 50 78 146 127 236 172 45 213)(5 101 51 79 147 128 237 173 46 214)(6 102 52 80 148 129 238 174 47 215)(7 103 53 81 149 130 239 175 48 216)(8 104 54 82 150 131 240 176 37 205)(9 105 55 83 151 132 229 177 38 206)(10 106 56 84 152 121 230 178 39 207)(11 107 57 73 153 122 231 179 40 208)(12 108 58 74 154 123 232 180 41 209)(13 182 142 66 119 96 201 25 224 163)(14 183 143 67 120 85 202 26 225 164)(15 184 144 68 109 86 203 27 226 165)(16 185 133 69 110 87 204 28 227 166)(17 186 134 70 111 88 193 29 228 167)(18 187 135 71 112 89 194 30 217 168)(19 188 136 72 113 90 195 31 218 157)(20 189 137 61 114 91 196 32 219 158)(21 190 138 62 115 92 197 33 220 159)(22 191 139 63 116 93 198 34 221 160)(23 192 140 64 117 94 199 35 222 161)(24 181 141 65 118 95 200 36 223 162)
(1 220 124 62)(2 63 125 221)(3 222 126 64)(4 65 127 223)(5 224 128 66)(6 67 129 225)(7 226 130 68)(8 69 131 227)(9 228 132 70)(10 71 121 217)(11 218 122 72)(12 61 123 219)(13 79 96 46)(14 47 85 80)(15 81 86 48)(16 37 87 82)(17 83 88 38)(18 39 89 84)(19 73 90 40)(20 41 91 74)(21 75 92 42)(22 43 93 76)(23 77 94 44)(24 45 95 78)(25 237 142 101)(26 102 143 238)(27 239 144 103)(28 104 133 240)(29 229 134 105)(30 106 135 230)(31 231 136 107)(32 108 137 232)(33 233 138 97)(34 98 139 234)(35 235 140 99)(36 100 141 236)(49 199 171 192)(50 181 172 200)(51 201 173 182)(52 183 174 202)(53 203 175 184)(54 185 176 204)(55 193 177 186)(56 187 178 194)(57 195 179 188)(58 189 180 196)(59 197 169 190)(60 191 170 198)(109 216 165 149)(110 150 166 205)(111 206 167 151)(112 152 168 207)(113 208 157 153)(114 154 158 209)(115 210 159 155)(116 156 160 211)(117 212 161 145)(118 146 162 213)(119 214 163 147)(120 148 164 215)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 76)(74 75)(77 84)(78 83)(79 82)(80 81)(85 86)(87 96)(88 95)(89 94)(90 93)(91 92)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 126)(122 125)(123 124)(127 132)(128 131)(129 130)(133 142)(134 141)(135 140)(136 139)(137 138)(143 144)(145 152)(146 151)(147 150)(148 149)(153 156)(154 155)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 186)(182 185)(183 184)(187 192)(188 191)(189 190)(193 200)(194 199)(195 198)(196 197)(201 204)(202 203)(205 214)(206 213)(207 212)(208 211)(209 210)(215 216)(217 222)(218 221)(219 220)(223 228)(224 227)(225 226)(229 236)(230 235)(231 234)(232 233)(237 240)(238 239)
G:=sub<Sym(240)| (1,97,59,75,155,124,233,169,42,210)(2,98,60,76,156,125,234,170,43,211)(3,99,49,77,145,126,235,171,44,212)(4,100,50,78,146,127,236,172,45,213)(5,101,51,79,147,128,237,173,46,214)(6,102,52,80,148,129,238,174,47,215)(7,103,53,81,149,130,239,175,48,216)(8,104,54,82,150,131,240,176,37,205)(9,105,55,83,151,132,229,177,38,206)(10,106,56,84,152,121,230,178,39,207)(11,107,57,73,153,122,231,179,40,208)(12,108,58,74,154,123,232,180,41,209)(13,182,142,66,119,96,201,25,224,163)(14,183,143,67,120,85,202,26,225,164)(15,184,144,68,109,86,203,27,226,165)(16,185,133,69,110,87,204,28,227,166)(17,186,134,70,111,88,193,29,228,167)(18,187,135,71,112,89,194,30,217,168)(19,188,136,72,113,90,195,31,218,157)(20,189,137,61,114,91,196,32,219,158)(21,190,138,62,115,92,197,33,220,159)(22,191,139,63,116,93,198,34,221,160)(23,192,140,64,117,94,199,35,222,161)(24,181,141,65,118,95,200,36,223,162), (1,220,124,62)(2,63,125,221)(3,222,126,64)(4,65,127,223)(5,224,128,66)(6,67,129,225)(7,226,130,68)(8,69,131,227)(9,228,132,70)(10,71,121,217)(11,218,122,72)(12,61,123,219)(13,79,96,46)(14,47,85,80)(15,81,86,48)(16,37,87,82)(17,83,88,38)(18,39,89,84)(19,73,90,40)(20,41,91,74)(21,75,92,42)(22,43,93,76)(23,77,94,44)(24,45,95,78)(25,237,142,101)(26,102,143,238)(27,239,144,103)(28,104,133,240)(29,229,134,105)(30,106,135,230)(31,231,136,107)(32,108,137,232)(33,233,138,97)(34,98,139,234)(35,235,140,99)(36,100,141,236)(49,199,171,192)(50,181,172,200)(51,201,173,182)(52,183,174,202)(53,203,175,184)(54,185,176,204)(55,193,177,186)(56,187,178,194)(57,195,179,188)(58,189,180,196)(59,197,169,190)(60,191,170,198)(109,216,165,149)(110,150,166,205)(111,206,167,151)(112,152,168,207)(113,208,157,153)(114,154,158,209)(115,210,159,155)(116,156,160,211)(117,212,161,145)(118,146,162,213)(119,214,163,147)(120,148,164,215), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81)(85,86)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,152)(146,151)(147,150)(148,149)(153,156)(154,155)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,186)(182,185)(183,184)(187,192)(188,191)(189,190)(193,200)(194,199)(195,198)(196,197)(201,204)(202,203)(205,214)(206,213)(207,212)(208,211)(209,210)(215,216)(217,222)(218,221)(219,220)(223,228)(224,227)(225,226)(229,236)(230,235)(231,234)(232,233)(237,240)(238,239)>;
G:=Group( (1,97,59,75,155,124,233,169,42,210)(2,98,60,76,156,125,234,170,43,211)(3,99,49,77,145,126,235,171,44,212)(4,100,50,78,146,127,236,172,45,213)(5,101,51,79,147,128,237,173,46,214)(6,102,52,80,148,129,238,174,47,215)(7,103,53,81,149,130,239,175,48,216)(8,104,54,82,150,131,240,176,37,205)(9,105,55,83,151,132,229,177,38,206)(10,106,56,84,152,121,230,178,39,207)(11,107,57,73,153,122,231,179,40,208)(12,108,58,74,154,123,232,180,41,209)(13,182,142,66,119,96,201,25,224,163)(14,183,143,67,120,85,202,26,225,164)(15,184,144,68,109,86,203,27,226,165)(16,185,133,69,110,87,204,28,227,166)(17,186,134,70,111,88,193,29,228,167)(18,187,135,71,112,89,194,30,217,168)(19,188,136,72,113,90,195,31,218,157)(20,189,137,61,114,91,196,32,219,158)(21,190,138,62,115,92,197,33,220,159)(22,191,139,63,116,93,198,34,221,160)(23,192,140,64,117,94,199,35,222,161)(24,181,141,65,118,95,200,36,223,162), (1,220,124,62)(2,63,125,221)(3,222,126,64)(4,65,127,223)(5,224,128,66)(6,67,129,225)(7,226,130,68)(8,69,131,227)(9,228,132,70)(10,71,121,217)(11,218,122,72)(12,61,123,219)(13,79,96,46)(14,47,85,80)(15,81,86,48)(16,37,87,82)(17,83,88,38)(18,39,89,84)(19,73,90,40)(20,41,91,74)(21,75,92,42)(22,43,93,76)(23,77,94,44)(24,45,95,78)(25,237,142,101)(26,102,143,238)(27,239,144,103)(28,104,133,240)(29,229,134,105)(30,106,135,230)(31,231,136,107)(32,108,137,232)(33,233,138,97)(34,98,139,234)(35,235,140,99)(36,100,141,236)(49,199,171,192)(50,181,172,200)(51,201,173,182)(52,183,174,202)(53,203,175,184)(54,185,176,204)(55,193,177,186)(56,187,178,194)(57,195,179,188)(58,189,180,196)(59,197,169,190)(60,191,170,198)(109,216,165,149)(110,150,166,205)(111,206,167,151)(112,152,168,207)(113,208,157,153)(114,154,158,209)(115,210,159,155)(116,156,160,211)(117,212,161,145)(118,146,162,213)(119,214,163,147)(120,148,164,215), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81)(85,86)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,142)(134,141)(135,140)(136,139)(137,138)(143,144)(145,152)(146,151)(147,150)(148,149)(153,156)(154,155)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,186)(182,185)(183,184)(187,192)(188,191)(189,190)(193,200)(194,199)(195,198)(196,197)(201,204)(202,203)(205,214)(206,213)(207,212)(208,211)(209,210)(215,216)(217,222)(218,221)(219,220)(223,228)(224,227)(225,226)(229,236)(230,235)(231,234)(232,233)(237,240)(238,239) );
G=PermutationGroup([[(1,97,59,75,155,124,233,169,42,210),(2,98,60,76,156,125,234,170,43,211),(3,99,49,77,145,126,235,171,44,212),(4,100,50,78,146,127,236,172,45,213),(5,101,51,79,147,128,237,173,46,214),(6,102,52,80,148,129,238,174,47,215),(7,103,53,81,149,130,239,175,48,216),(8,104,54,82,150,131,240,176,37,205),(9,105,55,83,151,132,229,177,38,206),(10,106,56,84,152,121,230,178,39,207),(11,107,57,73,153,122,231,179,40,208),(12,108,58,74,154,123,232,180,41,209),(13,182,142,66,119,96,201,25,224,163),(14,183,143,67,120,85,202,26,225,164),(15,184,144,68,109,86,203,27,226,165),(16,185,133,69,110,87,204,28,227,166),(17,186,134,70,111,88,193,29,228,167),(18,187,135,71,112,89,194,30,217,168),(19,188,136,72,113,90,195,31,218,157),(20,189,137,61,114,91,196,32,219,158),(21,190,138,62,115,92,197,33,220,159),(22,191,139,63,116,93,198,34,221,160),(23,192,140,64,117,94,199,35,222,161),(24,181,141,65,118,95,200,36,223,162)], [(1,220,124,62),(2,63,125,221),(3,222,126,64),(4,65,127,223),(5,224,128,66),(6,67,129,225),(7,226,130,68),(8,69,131,227),(9,228,132,70),(10,71,121,217),(11,218,122,72),(12,61,123,219),(13,79,96,46),(14,47,85,80),(15,81,86,48),(16,37,87,82),(17,83,88,38),(18,39,89,84),(19,73,90,40),(20,41,91,74),(21,75,92,42),(22,43,93,76),(23,77,94,44),(24,45,95,78),(25,237,142,101),(26,102,143,238),(27,239,144,103),(28,104,133,240),(29,229,134,105),(30,106,135,230),(31,231,136,107),(32,108,137,232),(33,233,138,97),(34,98,139,234),(35,235,140,99),(36,100,141,236),(49,199,171,192),(50,181,172,200),(51,201,173,182),(52,183,174,202),(53,203,175,184),(54,185,176,204),(55,193,177,186),(56,187,178,194),(57,195,179,188),(58,189,180,196),(59,197,169,190),(60,191,170,198),(109,216,165,149),(110,150,166,205),(111,206,167,151),(112,152,168,207),(113,208,157,153),(114,154,158,209),(115,210,159,155),(116,156,160,211),(117,212,161,145),(118,146,162,213),(119,214,163,147),(120,148,164,215)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,62),(63,72),(64,71),(65,70),(66,69),(67,68),(73,76),(74,75),(77,84),(78,83),(79,82),(80,81),(85,86),(87,96),(88,95),(89,94),(90,93),(91,92),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(121,126),(122,125),(123,124),(127,132),(128,131),(129,130),(133,142),(134,141),(135,140),(136,139),(137,138),(143,144),(145,152),(146,151),(147,150),(148,149),(153,156),(154,155),(157,160),(158,159),(161,168),(162,167),(163,166),(164,165),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,186),(182,185),(183,184),(187,192),(188,191),(189,190),(193,200),(194,199),(195,198),(196,197),(201,204),(202,203),(205,214),(206,213),(207,212),(208,211),(209,210),(215,216),(217,222),(218,221),(219,220),(223,228),(224,227),(225,226),(229,236),(230,235),(231,234),(232,233),(237,240),(238,239)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 60 | 2 | 4 | 10 | 10 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D12 | C5⋊D4 | S3×D4 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊D5 | D5×D12 | S3×C5⋊D4 |
| kernel | Dic5⋊D12 | D6⋊Dic5 | C3×C10.D4 | D30⋊3C4 | C2×S3×Dic5 | C2×C5⋊D12 | C10×D12 | C10.D4 | C3×Dic5 | S3×C10 | C2×D12 | C2×Dic5 | C2×C20 | C30 | C2×C12 | C22×S3 | Dic5 | D6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic5⋊D12 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 18 | 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 | 0 | 1 |
| 15 | 11 | 0 | 0 | 0 | 0 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 30 | 44 | 0 | 0 | 0 | 0 |
| 17 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 20 | 0 | 0 |
| 0 | 0 | 22 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 41 |
| 0 | 0 | 0 | 0 | 52 | 1 |
| 30 | 44 | 0 | 0 | 0 | 0 |
| 17 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 20 | 0 | 0 |
| 0 | 0 | 16 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 41 |
| 0 | 0 | 0 | 0 | 52 | 2 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,46,0,0,0,0,11,46,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[30,17,0,0,0,0,44,31,0,0,0,0,0,0,13,22,0,0,0,0,20,48,0,0,0,0,0,0,59,52,0,0,0,0,41,1],[30,17,0,0,0,0,44,31,0,0,0,0,0,0,13,16,0,0,0,0,20,48,0,0,0,0,0,0,59,52,0,0,0,0,41,2] >;
Dic5⋊D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes D_{12} % in TeX
G:=Group("Dic5:D12"); // GroupNames label
G:=SmallGroup(480,492);
// by ID
G=gap.SmallGroup(480,492);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,422,135,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations