metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.Dic5, Dic6.Dic5, C60.181C23, C3⋊D4.Dic5, C15⋊14(C8○D4), C20.52(C4×S3), C5⋊8(D12.C4), C4○D12.3D5, (C5×D12).4C4, (C2×C20).80D6, C5⋊2C8.25D6, C4.Dic5⋊5S3, C4.9(S3×Dic5), C60.130(C2×C4), (C4×S3).35D10, (C2×C12).81D10, C3⋊2(D4.Dic5), (C5×Dic6).4C4, D6.2(C2×Dic5), D6.Dic5⋊14C2, C12.11(C2×Dic5), C22.2(S3×Dic5), C6.4(C22×Dic5), (S3×C20).35C22, (C2×C60).214C22, C30.103(C22×C4), C20.178(C22×S3), C15⋊3C8.49C22, Dic3.2(C2×Dic5), C12.178(C22×D5), C4.151(C2×S3×D5), (S3×C5⋊2C8)⋊13C2, C2.6(C2×S3×Dic5), C10.112(S3×C2×C4), (C2×C15⋊3C8)⋊19C2, (C5×C3⋊D4).4C4, (C2×C10).31(C4×S3), (C5×C4○D12).6C2, (C2×C4).194(S3×D5), (C2×C6).5(C2×Dic5), (S3×C10).21(C2×C4), (C2×C30).100(C2×C4), (C3×C4.Dic5)⋊11C2, (C3×C5⋊2C8).25C22, (C5×Dic3).22(C2×C4), SmallGroup(480,364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.Dic5
G = < a,b,c,d | a12=b2=1, c10=a6, d2=c5, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c9 >
Subgroups: 380 in 124 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C30, C30, C8○D4, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C5×D4, C5×Q8, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C5×C4○D4, D12.C4, C3×C5⋊2C8, C15⋊3C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4.Dic5, S3×C5⋊2C8, D6.Dic5, C3×C4.Dic5, C2×C15⋊3C8, C5×C4○D12, D12.Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, Dic5, D10, C4×S3, C22×S3, C8○D4, C2×Dic5, C22×D5, S3×C2×C4, S3×D5, C22×Dic5, D12.C4, S3×Dic5, C2×S3×D5, D4.Dic5, C2×S3×Dic5, D12.Dic5
►On 240 points
Generators in S240
(1 239 193 212 58 93 11 229 183 202 48 83)(2 240 194 213 59 94 12 230 184 203 49 84)(3 221 195 214 60 95 13 231 185 204 50 85)(4 222 196 215 41 96 14 232 186 205 51 86)(5 223 197 216 42 97 15 233 187 206 52 87)(6 224 198 217 43 98 16 234 188 207 53 88)(7 225 199 218 44 99 17 235 189 208 54 89)(8 226 200 219 45 100 18 236 190 209 55 90)(9 227 181 220 46 81 19 237 191 210 56 91)(10 228 182 201 47 82 20 238 192 211 57 92)(21 112 150 179 127 75 31 102 160 169 137 65)(22 113 151 180 128 76 32 103 141 170 138 66)(23 114 152 161 129 77 33 104 142 171 139 67)(24 115 153 162 130 78 34 105 143 172 140 68)(25 116 154 163 131 79 35 106 144 173 121 69)(26 117 155 164 132 80 36 107 145 174 122 70)(27 118 156 165 133 61 37 108 146 175 123 71)(28 119 157 166 134 62 38 109 147 176 124 72)(29 120 158 167 135 63 39 110 148 177 125 73)(30 101 159 168 136 64 40 111 149 178 126 74)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 81)(20 82)(21 179)(22 180)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 167)(30 168)(31 169)(32 170)(33 171)(34 172)(35 173)(36 174)(37 175)(38 176)(39 177)(40 178)(41 232)(42 233)(43 234)(44 235)(45 236)(46 237)(47 238)(48 239)(49 240)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 229)(59 230)(60 231)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 121)(80 122)(101 159)(102 160)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(181 210)(182 211)(183 212)(184 213)(185 214)(186 215)(187 216)(188 217)(189 218)(190 219)(191 220)(192 201)(193 202)(194 203)(195 204)(196 205)(197 206)(198 207)(199 208)(200 209)
(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)
(1 143 6 148 11 153 16 158)(2 152 7 157 12 142 17 147)(3 141 8 146 13 151 18 156)(4 150 9 155 14 160 19 145)(5 159 10 144 15 149 20 154)(21 56 26 41 31 46 36 51)(22 45 27 50 32 55 37 60)(23 54 28 59 33 44 38 49)(24 43 29 48 34 53 39 58)(25 52 30 57 35 42 40 47)(61 204 66 209 71 214 76 219)(62 213 67 218 72 203 77 208)(63 202 68 207 73 212 78 217)(64 211 69 216 74 201 79 206)(65 220 70 205 75 210 80 215)(81 117 86 102 91 107 96 112)(82 106 87 111 92 116 97 101)(83 115 88 120 93 105 98 110)(84 104 89 109 94 114 99 119)(85 113 90 118 95 103 100 108)(121 187 126 192 131 197 136 182)(122 196 127 181 132 186 137 191)(123 185 128 190 133 195 138 200)(124 194 129 199 134 184 139 189)(125 183 130 188 135 193 140 198)(161 235 166 240 171 225 176 230)(162 224 167 229 172 234 177 239)(163 233 168 238 173 223 178 228)(164 222 169 227 174 232 179 237)(165 231 170 236 175 221 180 226)
G:=sub<Sym(240)| (1,239,193,212,58,93,11,229,183,202,48,83)(2,240,194,213,59,94,12,230,184,203,49,84)(3,221,195,214,60,95,13,231,185,204,50,85)(4,222,196,215,41,96,14,232,186,205,51,86)(5,223,197,216,42,97,15,233,187,206,52,87)(6,224,198,217,43,98,16,234,188,207,53,88)(7,225,199,218,44,99,17,235,189,208,54,89)(8,226,200,219,45,100,18,236,190,209,55,90)(9,227,181,220,46,81,19,237,191,210,56,91)(10,228,182,201,47,82,20,238,192,211,57,92)(21,112,150,179,127,75,31,102,160,169,137,65)(22,113,151,180,128,76,32,103,141,170,138,66)(23,114,152,161,129,77,33,104,142,171,139,67)(24,115,153,162,130,78,34,105,143,172,140,68)(25,116,154,163,131,79,35,106,144,173,121,69)(26,117,155,164,132,80,36,107,145,174,122,70)(27,118,156,165,133,61,37,108,146,175,123,71)(28,119,157,166,134,62,38,109,147,176,124,72)(29,120,158,167,135,63,39,110,148,177,125,73)(30,101,159,168,136,64,40,111,149,178,126,74), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,179)(22,180)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,169)(32,170)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,177)(40,178)(41,232)(42,233)(43,234)(44,235)(45,236)(46,237)(47,238)(48,239)(49,240)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,121)(80,122)(101,159)(102,160)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,209), (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), (1,143,6,148,11,153,16,158)(2,152,7,157,12,142,17,147)(3,141,8,146,13,151,18,156)(4,150,9,155,14,160,19,145)(5,159,10,144,15,149,20,154)(21,56,26,41,31,46,36,51)(22,45,27,50,32,55,37,60)(23,54,28,59,33,44,38,49)(24,43,29,48,34,53,39,58)(25,52,30,57,35,42,40,47)(61,204,66,209,71,214,76,219)(62,213,67,218,72,203,77,208)(63,202,68,207,73,212,78,217)(64,211,69,216,74,201,79,206)(65,220,70,205,75,210,80,215)(81,117,86,102,91,107,96,112)(82,106,87,111,92,116,97,101)(83,115,88,120,93,105,98,110)(84,104,89,109,94,114,99,119)(85,113,90,118,95,103,100,108)(121,187,126,192,131,197,136,182)(122,196,127,181,132,186,137,191)(123,185,128,190,133,195,138,200)(124,194,129,199,134,184,139,189)(125,183,130,188,135,193,140,198)(161,235,166,240,171,225,176,230)(162,224,167,229,172,234,177,239)(163,233,168,238,173,223,178,228)(164,222,169,227,174,232,179,237)(165,231,170,236,175,221,180,226)>;
G:=Group( (1,239,193,212,58,93,11,229,183,202,48,83)(2,240,194,213,59,94,12,230,184,203,49,84)(3,221,195,214,60,95,13,231,185,204,50,85)(4,222,196,215,41,96,14,232,186,205,51,86)(5,223,197,216,42,97,15,233,187,206,52,87)(6,224,198,217,43,98,16,234,188,207,53,88)(7,225,199,218,44,99,17,235,189,208,54,89)(8,226,200,219,45,100,18,236,190,209,55,90)(9,227,181,220,46,81,19,237,191,210,56,91)(10,228,182,201,47,82,20,238,192,211,57,92)(21,112,150,179,127,75,31,102,160,169,137,65)(22,113,151,180,128,76,32,103,141,170,138,66)(23,114,152,161,129,77,33,104,142,171,139,67)(24,115,153,162,130,78,34,105,143,172,140,68)(25,116,154,163,131,79,35,106,144,173,121,69)(26,117,155,164,132,80,36,107,145,174,122,70)(27,118,156,165,133,61,37,108,146,175,123,71)(28,119,157,166,134,62,38,109,147,176,124,72)(29,120,158,167,135,63,39,110,148,177,125,73)(30,101,159,168,136,64,40,111,149,178,126,74), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,179)(22,180)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,169)(32,170)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,177)(40,178)(41,232)(42,233)(43,234)(44,235)(45,236)(46,237)(47,238)(48,239)(49,240)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,121)(80,122)(101,159)(102,160)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)(199,208)(200,209), (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), (1,143,6,148,11,153,16,158)(2,152,7,157,12,142,17,147)(3,141,8,146,13,151,18,156)(4,150,9,155,14,160,19,145)(5,159,10,144,15,149,20,154)(21,56,26,41,31,46,36,51)(22,45,27,50,32,55,37,60)(23,54,28,59,33,44,38,49)(24,43,29,48,34,53,39,58)(25,52,30,57,35,42,40,47)(61,204,66,209,71,214,76,219)(62,213,67,218,72,203,77,208)(63,202,68,207,73,212,78,217)(64,211,69,216,74,201,79,206)(65,220,70,205,75,210,80,215)(81,117,86,102,91,107,96,112)(82,106,87,111,92,116,97,101)(83,115,88,120,93,105,98,110)(84,104,89,109,94,114,99,119)(85,113,90,118,95,103,100,108)(121,187,126,192,131,197,136,182)(122,196,127,181,132,186,137,191)(123,185,128,190,133,195,138,200)(124,194,129,199,134,184,139,189)(125,183,130,188,135,193,140,198)(161,235,166,240,171,225,176,230)(162,224,167,229,172,234,177,239)(163,233,168,238,173,223,178,228)(164,222,169,227,174,232,179,237)(165,231,170,236,175,221,180,226) );
G=PermutationGroup([[(1,239,193,212,58,93,11,229,183,202,48,83),(2,240,194,213,59,94,12,230,184,203,49,84),(3,221,195,214,60,95,13,231,185,204,50,85),(4,222,196,215,41,96,14,232,186,205,51,86),(5,223,197,216,42,97,15,233,187,206,52,87),(6,224,198,217,43,98,16,234,188,207,53,88),(7,225,199,218,44,99,17,235,189,208,54,89),(8,226,200,219,45,100,18,236,190,209,55,90),(9,227,181,220,46,81,19,237,191,210,56,91),(10,228,182,201,47,82,20,238,192,211,57,92),(21,112,150,179,127,75,31,102,160,169,137,65),(22,113,151,180,128,76,32,103,141,170,138,66),(23,114,152,161,129,77,33,104,142,171,139,67),(24,115,153,162,130,78,34,105,143,172,140,68),(25,116,154,163,131,79,35,106,144,173,121,69),(26,117,155,164,132,80,36,107,145,174,122,70),(27,118,156,165,133,61,37,108,146,175,123,71),(28,119,157,166,134,62,38,109,147,176,124,72),(29,120,158,167,135,63,39,110,148,177,125,73),(30,101,159,168,136,64,40,111,149,178,126,74)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,81),(20,82),(21,179),(22,180),(23,161),(24,162),(25,163),(26,164),(27,165),(28,166),(29,167),(30,168),(31,169),(32,170),(33,171),(34,172),(35,173),(36,174),(37,175),(38,176),(39,177),(40,178),(41,232),(42,233),(43,234),(44,235),(45,236),(46,237),(47,238),(48,239),(49,240),(50,221),(51,222),(52,223),(53,224),(54,225),(55,226),(56,227),(57,228),(58,229),(59,230),(60,231),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,121),(80,122),(101,159),(102,160),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148),(111,149),(112,150),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158),(181,210),(182,211),(183,212),(184,213),(185,214),(186,215),(187,216),(188,217),(189,218),(190,219),(191,220),(192,201),(193,202),(194,203),(195,204),(196,205),(197,206),(198,207),(199,208),(200,209)], [(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)], [(1,143,6,148,11,153,16,158),(2,152,7,157,12,142,17,147),(3,141,8,146,13,151,18,156),(4,150,9,155,14,160,19,145),(5,159,10,144,15,149,20,154),(21,56,26,41,31,46,36,51),(22,45,27,50,32,55,37,60),(23,54,28,59,33,44,38,49),(24,43,29,48,34,53,39,58),(25,52,30,57,35,42,40,47),(61,204,66,209,71,214,76,219),(62,213,67,218,72,203,77,208),(63,202,68,207,73,212,78,217),(64,211,69,216,74,201,79,206),(65,220,70,205,75,210,80,215),(81,117,86,102,91,107,96,112),(82,106,87,111,92,116,97,101),(83,115,88,120,93,105,98,110),(84,104,89,109,94,114,99,119),(85,113,90,118,95,103,100,108),(121,187,126,192,131,197,136,182),(122,196,127,181,132,186,137,191),(123,185,128,190,133,195,138,200),(124,194,129,199,134,184,139,189),(125,183,130,188,135,193,140,198),(161,235,166,240,171,225,176,230),(162,224,167,229,172,234,177,239),(163,233,168,238,173,223,178,228),(164,222,169,227,174,232,179,237),(165,231,170,236,175,221,180,226)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | - | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | Dic5 | D10 | Dic5 | Dic5 | D10 | C4×S3 | C4×S3 | C8○D4 | S3×D5 | D12.C4 | S3×Dic5 | C2×S3×D5 | S3×Dic5 | D4.Dic5 | D12.Dic5 |
| kernel | D12.Dic5 | S3×C5⋊2C8 | D6.Dic5 | C3×C4.Dic5 | C2×C15⋊3C8 | C5×C4○D12 | C5×Dic6 | C5×D12 | C5×C3⋊D4 | C4.Dic5 | C4○D12 | C5⋊2C8 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C20 | C2×C10 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D12.Dic5 ►in GL6(𝔽241)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 20 | 205 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 211 | 0 | 0 |
| 0 | 0 | 0 | 0 | 217 | 3 |
| 0 | 0 | 0 | 0 | 210 | 24 |
G:=sub<GL(6,GF(241))| [1,240,0,0,0,0,1,0,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,1,240,0,0,0,0,0,0,0,64,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,87,20,0,0,0,0,0,205],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,30,0,0,0,0,0,0,211,0,0,0,0,0,0,217,210,0,0,0,0,3,24] >;
D12.Dic5 in GAP, Magma, Sage, TeX
D_{12}.{\rm Dic}_5 % in TeX
G:=Group("D12.Dic5"); // GroupNames label
G:=SmallGroup(480,364);
// by ID
G=gap.SmallGroup(480,364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^10=a^6,d^2=c^5,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^9>;
// generators/relations