metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.2Dic5, C60.179C23, Dic6.2Dic5, C5⋊8(C8○D12), C15⋊13(C8○D4), C20.51(C4×S3), C4○D12.6D5, (C5×D12).3C4, (C2×C20).79D6, C5⋊2C8.38D6, C4.4(S3×Dic5), C60.115(C2×C4), (C4×S3).34D10, C3⋊1(D4.Dic5), D6.1(C2×Dic5), (C5×Dic6).3C4, C60.7C4⋊13C2, C3⋊D4.2Dic5, (C2×C12).316D10, D6.Dic5⋊12C2, (C2×C60).47C22, C12.19(C2×Dic5), C6.2(C22×Dic5), C22.1(S3×Dic5), (S3×C20).34C22, C20.176(C22×S3), C30.101(C22×C4), C15⋊3C8.30C22, Dic3.1(C2×Dic5), C12.176(C22×D5), (C6×C5⋊2C8)⋊1C2, (C2×C5⋊2C8)⋊3S3, C4.149(C2×S3×D5), (S3×C5⋊2C8)⋊11C2, C2.4(C2×S3×Dic5), C10.110(S3×C2×C4), (C5×C3⋊D4).3C4, (C2×C10).30(C4×S3), (C2×C30).98(C2×C4), (C5×C4○D12).3C2, (C2×C4).143(S3×D5), (S3×C10).20(C2×C4), (C2×C6).15(C2×Dic5), (C3×C5⋊2C8).43C22, (C5×Dic3).21(C2×C4), SmallGroup(480,362)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2Dic5
G = < a,b,c,d | a12=b2=1, c10=a6, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 380 in 124 conjugacy classes, 60 normal (38 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, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C5×C4○D4, C8○D12, C3×C5⋊2C8, C15⋊3C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4.Dic5, S3×C5⋊2C8, D6.Dic5, C6×C5⋊2C8, C60.7C4, C5×C4○D12, D12.2Dic5
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, C8○D12, S3×Dic5, C2×S3×D5, D4.Dic5, C2×S3×Dic5, D12.2Dic5
►On 240 points
Generators in S240
(1 45 211 111 27 82 11 55 201 101 37 92)(2 46 212 112 28 83 12 56 202 102 38 93)(3 47 213 113 29 84 13 57 203 103 39 94)(4 48 214 114 30 85 14 58 204 104 40 95)(5 49 215 115 31 86 15 59 205 105 21 96)(6 50 216 116 32 87 16 60 206 106 22 97)(7 51 217 117 33 88 17 41 207 107 23 98)(8 52 218 118 34 89 18 42 208 108 24 99)(9 53 219 119 35 90 19 43 209 109 25 100)(10 54 220 120 36 91 20 44 210 110 26 81)(61 194 223 156 165 126 71 184 233 146 175 136)(62 195 224 157 166 127 72 185 234 147 176 137)(63 196 225 158 167 128 73 186 235 148 177 138)(64 197 226 159 168 129 74 187 236 149 178 139)(65 198 227 160 169 130 75 188 237 150 179 140)(66 199 228 141 170 131 76 189 238 151 180 121)(67 200 229 142 171 132 77 190 239 152 161 122)(68 181 230 143 172 133 78 191 240 153 162 123)(69 182 231 144 173 134 79 192 221 154 163 124)(70 183 232 145 174 135 80 193 222 155 164 125)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(61 156)(62 157)(63 158)(64 159)(65 160)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(101 211)(102 212)(103 213)(104 214)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 201)(112 202)(113 203)(114 204)(115 205)(116 206)(117 207)(118 208)(119 209)(120 210)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(181 230)(182 231)(183 232)(184 233)(185 234)(186 235)(187 236)(188 237)(189 238)(190 239)(191 240)(192 221)(193 222)(194 223)(195 224)(196 225)(197 226)(198 227)(199 228)(200 229)
(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)
(1 230 6 235 11 240 16 225)(2 239 7 224 12 229 17 234)(3 228 8 233 13 238 18 223)(4 237 9 222 14 227 19 232)(5 226 10 231 15 236 20 221)(21 64 26 69 31 74 36 79)(22 73 27 78 32 63 37 68)(23 62 28 67 33 72 38 77)(24 71 29 76 34 61 39 66)(25 80 30 65 35 70 40 75)(41 147 46 152 51 157 56 142)(42 156 47 141 52 146 57 151)(43 145 48 150 53 155 58 160)(44 154 49 159 54 144 59 149)(45 143 50 148 55 153 60 158)(81 182 86 187 91 192 96 197)(82 191 87 196 92 181 97 186)(83 200 88 185 93 190 98 195)(84 189 89 194 94 199 99 184)(85 198 90 183 95 188 100 193)(101 123 106 128 111 133 116 138)(102 132 107 137 112 122 117 127)(103 121 108 126 113 131 118 136)(104 130 109 135 114 140 119 125)(105 139 110 124 115 129 120 134)(161 217 166 202 171 207 176 212)(162 206 167 211 172 216 177 201)(163 215 168 220 173 205 178 210)(164 204 169 209 174 214 179 219)(165 213 170 218 175 203 180 208)
G:=sub<Sym(240)| (1,45,211,111,27,82,11,55,201,101,37,92)(2,46,212,112,28,83,12,56,202,102,38,93)(3,47,213,113,29,84,13,57,203,103,39,94)(4,48,214,114,30,85,14,58,204,104,40,95)(5,49,215,115,31,86,15,59,205,105,21,96)(6,50,216,116,32,87,16,60,206,106,22,97)(7,51,217,117,33,88,17,41,207,107,23,98)(8,52,218,118,34,89,18,42,208,108,24,99)(9,53,219,119,35,90,19,43,209,109,25,100)(10,54,220,120,36,91,20,44,210,110,26,81)(61,194,223,156,165,126,71,184,233,146,175,136)(62,195,224,157,166,127,72,185,234,147,176,137)(63,196,225,158,167,128,73,186,235,148,177,138)(64,197,226,159,168,129,74,187,236,149,178,139)(65,198,227,160,169,130,75,188,237,150,179,140)(66,199,228,141,170,131,76,189,238,151,180,121)(67,200,229,142,171,132,77,190,239,152,161,122)(68,181,230,143,172,133,78,191,240,153,162,123)(69,182,231,144,173,134,79,192,221,154,163,124)(70,183,232,145,174,135,80,193,222,155,164,125), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(61,156)(62,157)(63,158)(64,159)(65,160)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,201)(112,202)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229), (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), (1,230,6,235,11,240,16,225)(2,239,7,224,12,229,17,234)(3,228,8,233,13,238,18,223)(4,237,9,222,14,227,19,232)(5,226,10,231,15,236,20,221)(21,64,26,69,31,74,36,79)(22,73,27,78,32,63,37,68)(23,62,28,67,33,72,38,77)(24,71,29,76,34,61,39,66)(25,80,30,65,35,70,40,75)(41,147,46,152,51,157,56,142)(42,156,47,141,52,146,57,151)(43,145,48,150,53,155,58,160)(44,154,49,159,54,144,59,149)(45,143,50,148,55,153,60,158)(81,182,86,187,91,192,96,197)(82,191,87,196,92,181,97,186)(83,200,88,185,93,190,98,195)(84,189,89,194,94,199,99,184)(85,198,90,183,95,188,100,193)(101,123,106,128,111,133,116,138)(102,132,107,137,112,122,117,127)(103,121,108,126,113,131,118,136)(104,130,109,135,114,140,119,125)(105,139,110,124,115,129,120,134)(161,217,166,202,171,207,176,212)(162,206,167,211,172,216,177,201)(163,215,168,220,173,205,178,210)(164,204,169,209,174,214,179,219)(165,213,170,218,175,203,180,208)>;
G:=Group( (1,45,211,111,27,82,11,55,201,101,37,92)(2,46,212,112,28,83,12,56,202,102,38,93)(3,47,213,113,29,84,13,57,203,103,39,94)(4,48,214,114,30,85,14,58,204,104,40,95)(5,49,215,115,31,86,15,59,205,105,21,96)(6,50,216,116,32,87,16,60,206,106,22,97)(7,51,217,117,33,88,17,41,207,107,23,98)(8,52,218,118,34,89,18,42,208,108,24,99)(9,53,219,119,35,90,19,43,209,109,25,100)(10,54,220,120,36,91,20,44,210,110,26,81)(61,194,223,156,165,126,71,184,233,146,175,136)(62,195,224,157,166,127,72,185,234,147,176,137)(63,196,225,158,167,128,73,186,235,148,177,138)(64,197,226,159,168,129,74,187,236,149,178,139)(65,198,227,160,169,130,75,188,237,150,179,140)(66,199,228,141,170,131,76,189,238,151,180,121)(67,200,229,142,171,132,77,190,239,152,161,122)(68,181,230,143,172,133,78,191,240,153,162,123)(69,182,231,144,173,134,79,192,221,154,163,124)(70,183,232,145,174,135,80,193,222,155,164,125), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(61,156)(62,157)(63,158)(64,159)(65,160)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,201)(112,202)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229), (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), (1,230,6,235,11,240,16,225)(2,239,7,224,12,229,17,234)(3,228,8,233,13,238,18,223)(4,237,9,222,14,227,19,232)(5,226,10,231,15,236,20,221)(21,64,26,69,31,74,36,79)(22,73,27,78,32,63,37,68)(23,62,28,67,33,72,38,77)(24,71,29,76,34,61,39,66)(25,80,30,65,35,70,40,75)(41,147,46,152,51,157,56,142)(42,156,47,141,52,146,57,151)(43,145,48,150,53,155,58,160)(44,154,49,159,54,144,59,149)(45,143,50,148,55,153,60,158)(81,182,86,187,91,192,96,197)(82,191,87,196,92,181,97,186)(83,200,88,185,93,190,98,195)(84,189,89,194,94,199,99,184)(85,198,90,183,95,188,100,193)(101,123,106,128,111,133,116,138)(102,132,107,137,112,122,117,127)(103,121,108,126,113,131,118,136)(104,130,109,135,114,140,119,125)(105,139,110,124,115,129,120,134)(161,217,166,202,171,207,176,212)(162,206,167,211,172,216,177,201)(163,215,168,220,173,205,178,210)(164,204,169,209,174,214,179,219)(165,213,170,218,175,203,180,208) );
G=PermutationGroup([[(1,45,211,111,27,82,11,55,201,101,37,92),(2,46,212,112,28,83,12,56,202,102,38,93),(3,47,213,113,29,84,13,57,203,103,39,94),(4,48,214,114,30,85,14,58,204,104,40,95),(5,49,215,115,31,86,15,59,205,105,21,96),(6,50,216,116,32,87,16,60,206,106,22,97),(7,51,217,117,33,88,17,41,207,107,23,98),(8,52,218,118,34,89,18,42,208,108,24,99),(9,53,219,119,35,90,19,43,209,109,25,100),(10,54,220,120,36,91,20,44,210,110,26,81),(61,194,223,156,165,126,71,184,233,146,175,136),(62,195,224,157,166,127,72,185,234,147,176,137),(63,196,225,158,167,128,73,186,235,148,177,138),(64,197,226,159,168,129,74,187,236,149,178,139),(65,198,227,160,169,130,75,188,237,150,179,140),(66,199,228,141,170,131,76,189,238,151,180,121),(67,200,229,142,171,132,77,190,239,152,161,122),(68,181,230,143,172,133,78,191,240,153,162,123),(69,182,231,144,173,134,79,192,221,154,163,124),(70,183,232,145,174,135,80,193,222,155,164,125)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(61,156),(62,157),(63,158),(64,159),(65,160),(66,141),(67,142),(68,143),(69,144),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(101,211),(102,212),(103,213),(104,214),(105,215),(106,216),(107,217),(108,218),(109,219),(110,220),(111,201),(112,202),(113,203),(114,204),(115,205),(116,206),(117,207),(118,208),(119,209),(120,210),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,180),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(181,230),(182,231),(183,232),(184,233),(185,234),(186,235),(187,236),(188,237),(189,238),(190,239),(191,240),(192,221),(193,222),(194,223),(195,224),(196,225),(197,226),(198,227),(199,228),(200,229)], [(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)], [(1,230,6,235,11,240,16,225),(2,239,7,224,12,229,17,234),(3,228,8,233,13,238,18,223),(4,237,9,222,14,227,19,232),(5,226,10,231,15,236,20,221),(21,64,26,69,31,74,36,79),(22,73,27,78,32,63,37,68),(23,62,28,67,33,72,38,77),(24,71,29,76,34,61,39,66),(25,80,30,65,35,70,40,75),(41,147,46,152,51,157,56,142),(42,156,47,141,52,146,57,151),(43,145,48,150,53,155,58,160),(44,154,49,159,54,144,59,149),(45,143,50,148,55,153,60,158),(81,182,86,187,91,192,96,197),(82,191,87,196,92,181,97,186),(83,200,88,185,93,190,98,195),(84,189,89,194,94,199,99,184),(85,198,90,183,95,188,100,193),(101,123,106,128,111,133,116,138),(102,132,107,137,112,122,117,127),(103,121,108,126,113,131,118,136),(104,130,109,135,114,140,119,125),(105,139,110,124,115,129,120,134),(161,217,166,202,171,207,176,212),(162,206,167,211,172,216,177,201),(163,215,168,220,173,205,178,210),(164,204,169,209,174,214,179,219),(165,213,170,218,175,203,180,208)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C8○D12 | S3×D5 | S3×Dic5 | C2×S3×D5 | S3×Dic5 | D4.Dic5 | D12.2Dic5 |
| kernel | D12.2Dic5 | S3×C5⋊2C8 | D6.Dic5 | C6×C5⋊2C8 | C60.7C4 | C5×C4○D12 | C5×Dic6 | C5×D12 | C5×C3⋊D4 | C2×C5⋊2C8 | C4○D12 | C5⋊2C8 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C20 | C2×C10 | C15 | C5 | C2×C4 | 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 | 8 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D12.2Dic5 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 142 | 99 |
| 0 | 0 | 142 | 43 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 142 | 99 |
| 0 | 0 | 198 | 99 |
| 205 | 0 | 0 | 0 |
| 234 | 87 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 82 | 74 | 0 | 0 |
| 199 | 159 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,142,142,0,0,99,43],[240,0,0,0,0,240,0,0,0,0,142,198,0,0,99,99],[205,234,0,0,0,87,0,0,0,0,64,0,0,0,0,64],[82,199,0,0,74,159,0,0,0,0,8,0,0,0,0,8] >;
D12.2Dic5 in GAP, Magma, Sage, TeX
D_{12}._2{\rm Dic}_5 % in TeX
G:=Group("D12.2Dic5"); // GroupNames label
G:=SmallGroup(480,362);
// by ID
G=gap.SmallGroup(480,362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,80,1356,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations