metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.2F5, Dic30.2C4, C5⋊C8.4D6, D5⋊C8⋊2S3, C4.6(S3×F5), C5⋊2(C8○D12), C15⋊1(C8○D4), D6.F5⋊1C2, C3⋊1(D4.F5), D6.1(C2×F5), C20.11(C4×S3), C60.13(C2×C4), D10.1(C4×S3), (C4×D5).33D6, (C5×D12).2C4, C15⋊D4.1C4, C12.27(C2×F5), C12.F5⋊3C2, C6.6(C22×F5), C30.6(C22×C4), D12⋊5D5.6C2, C15⋊C8.1C22, Dic15.3(C2×C4), (D5×C12).43C22, (S3×Dic5).5C22, Dic5.26(C22×S3), (C3×Dic5).24C23, (S3×C5⋊C8)⋊1C2, C10.6(S3×C2×C4), (C3×D5⋊C8)⋊2C2, C2.10(C2×S3×F5), (C3×C5⋊C8).4C22, (S3×C10).1(C2×C4), (C6×D5).15(C2×C4), SmallGroup(480,987)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2F5
G = < a,b,c,d | a12=b2=c5=1, d4=a6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 548 in 124 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, C8○D4, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, D4⋊2D5, C8○D12, C3×C5⋊C8, C15⋊C8, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, D4.F5, S3×C5⋊C8, D6.F5, C3×D5⋊C8, C12.F5, D12⋊5D5, D12.2F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, F5, C4×S3, C22×S3, C8○D4, C2×F5, S3×C2×C4, C22×F5, C8○D12, S3×F5, D4.F5, C2×S3×F5, D12.2F5
►On 240 points
Generators in S240
(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 3)(4 12)(5 11)(6 10)(7 9)(13 17)(14 16)(18 24)(19 23)(20 22)(25 27)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 48)(49 59)(50 58)(51 57)(52 56)(53 55)(61 65)(62 64)(66 72)(67 71)(68 70)(73 81)(74 80)(75 79)(76 78)(82 84)(85 95)(86 94)(87 93)(88 92)(89 91)(97 99)(100 108)(101 107)(102 106)(103 105)(109 115)(110 114)(111 113)(116 120)(117 119)(121 129)(122 128)(123 127)(124 126)(130 132)(133 143)(134 142)(135 141)(136 140)(137 139)(146 156)(147 155)(148 154)(149 153)(150 152)(157 163)(158 162)(159 161)(164 168)(165 167)(169 179)(170 178)(171 177)(172 176)(173 175)(181 187)(182 186)(183 185)(188 192)(189 191)(193 199)(194 198)(195 197)(200 204)(201 203)(206 216)(207 215)(208 214)(209 213)(210 212)(217 219)(220 228)(221 227)(222 226)(223 225)(229 235)(230 234)(231 233)(236 240)(237 239)
(1 130 40 183 111)(2 131 41 184 112)(3 132 42 185 113)(4 121 43 186 114)(5 122 44 187 115)(6 123 45 188 116)(7 124 46 189 117)(8 125 47 190 118)(9 126 48 191 119)(10 127 37 192 120)(11 128 38 181 109)(12 129 39 182 110)(13 164 215 236 67)(14 165 216 237 68)(15 166 205 238 69)(16 167 206 239 70)(17 168 207 240 71)(18 157 208 229 72)(19 158 209 230 61)(20 159 210 231 62)(21 160 211 232 63)(22 161 212 233 64)(23 162 213 234 65)(24 163 214 235 66)(25 173 217 82 95)(26 174 218 83 96)(27 175 219 84 85)(28 176 220 73 86)(29 177 221 74 87)(30 178 222 75 88)(31 179 223 76 89)(32 180 224 77 90)(33 169 225 78 91)(34 170 226 79 92)(35 171 227 80 93)(36 172 228 81 94)(49 197 105 139 152)(50 198 106 140 153)(51 199 107 141 154)(52 200 108 142 155)(53 201 97 143 156)(54 202 98 144 145)(55 203 99 133 146)(56 204 100 134 147)(57 193 101 135 148)(58 194 102 136 149)(59 195 103 137 150)(60 196 104 138 151)
(1 25 143 165 7 31 137 159)(2 26 144 166 8 32 138 160)(3 27 133 167 9 33 139 161)(4 28 134 168 10 34 140 162)(5 29 135 157 11 35 141 163)(6 30 136 158 12 36 142 164)(13 188 178 58 19 182 172 52)(14 189 179 59 20 183 173 53)(15 190 180 60 21 184 174 54)(16 191 169 49 22 185 175 55)(17 192 170 50 23 186 176 56)(18 181 171 51 24 187 177 57)(37 92 198 213 43 86 204 207)(38 93 199 214 44 87 193 208)(39 94 200 215 45 88 194 209)(40 95 201 216 46 89 195 210)(41 96 202 205 47 90 196 211)(42 85 203 206 48 91 197 212)(61 129 228 108 67 123 222 102)(62 130 217 97 68 124 223 103)(63 131 218 98 69 125 224 104)(64 132 219 99 70 126 225 105)(65 121 220 100 71 127 226 106)(66 122 221 101 72 128 227 107)(73 147 240 120 79 153 234 114)(74 148 229 109 80 154 235 115)(75 149 230 110 81 155 236 116)(76 150 231 111 82 156 237 117)(77 151 232 112 83 145 238 118)(78 152 233 113 84 146 239 119)
G:=sub<Sym(240)| (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,72)(67,71)(68,70)(73,81)(74,80)(75,79)(76,78)(82,84)(85,95)(86,94)(87,93)(88,92)(89,91)(97,99)(100,108)(101,107)(102,106)(103,105)(109,115)(110,114)(111,113)(116,120)(117,119)(121,129)(122,128)(123,127)(124,126)(130,132)(133,143)(134,142)(135,141)(136,140)(137,139)(146,156)(147,155)(148,154)(149,153)(150,152)(157,163)(158,162)(159,161)(164,168)(165,167)(169,179)(170,178)(171,177)(172,176)(173,175)(181,187)(182,186)(183,185)(188,192)(189,191)(193,199)(194,198)(195,197)(200,204)(201,203)(206,216)(207,215)(208,214)(209,213)(210,212)(217,219)(220,228)(221,227)(222,226)(223,225)(229,235)(230,234)(231,233)(236,240)(237,239), (1,130,40,183,111)(2,131,41,184,112)(3,132,42,185,113)(4,121,43,186,114)(5,122,44,187,115)(6,123,45,188,116)(7,124,46,189,117)(8,125,47,190,118)(9,126,48,191,119)(10,127,37,192,120)(11,128,38,181,109)(12,129,39,182,110)(13,164,215,236,67)(14,165,216,237,68)(15,166,205,238,69)(16,167,206,239,70)(17,168,207,240,71)(18,157,208,229,72)(19,158,209,230,61)(20,159,210,231,62)(21,160,211,232,63)(22,161,212,233,64)(23,162,213,234,65)(24,163,214,235,66)(25,173,217,82,95)(26,174,218,83,96)(27,175,219,84,85)(28,176,220,73,86)(29,177,221,74,87)(30,178,222,75,88)(31,179,223,76,89)(32,180,224,77,90)(33,169,225,78,91)(34,170,226,79,92)(35,171,227,80,93)(36,172,228,81,94)(49,197,105,139,152)(50,198,106,140,153)(51,199,107,141,154)(52,200,108,142,155)(53,201,97,143,156)(54,202,98,144,145)(55,203,99,133,146)(56,204,100,134,147)(57,193,101,135,148)(58,194,102,136,149)(59,195,103,137,150)(60,196,104,138,151), (1,25,143,165,7,31,137,159)(2,26,144,166,8,32,138,160)(3,27,133,167,9,33,139,161)(4,28,134,168,10,34,140,162)(5,29,135,157,11,35,141,163)(6,30,136,158,12,36,142,164)(13,188,178,58,19,182,172,52)(14,189,179,59,20,183,173,53)(15,190,180,60,21,184,174,54)(16,191,169,49,22,185,175,55)(17,192,170,50,23,186,176,56)(18,181,171,51,24,187,177,57)(37,92,198,213,43,86,204,207)(38,93,199,214,44,87,193,208)(39,94,200,215,45,88,194,209)(40,95,201,216,46,89,195,210)(41,96,202,205,47,90,196,211)(42,85,203,206,48,91,197,212)(61,129,228,108,67,123,222,102)(62,130,217,97,68,124,223,103)(63,131,218,98,69,125,224,104)(64,132,219,99,70,126,225,105)(65,121,220,100,71,127,226,106)(66,122,221,101,72,128,227,107)(73,147,240,120,79,153,234,114)(74,148,229,109,80,154,235,115)(75,149,230,110,81,155,236,116)(76,150,231,111,82,156,237,117)(77,151,232,112,83,145,238,118)(78,152,233,113,84,146,239,119)>;
G:=Group( (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,72)(67,71)(68,70)(73,81)(74,80)(75,79)(76,78)(82,84)(85,95)(86,94)(87,93)(88,92)(89,91)(97,99)(100,108)(101,107)(102,106)(103,105)(109,115)(110,114)(111,113)(116,120)(117,119)(121,129)(122,128)(123,127)(124,126)(130,132)(133,143)(134,142)(135,141)(136,140)(137,139)(146,156)(147,155)(148,154)(149,153)(150,152)(157,163)(158,162)(159,161)(164,168)(165,167)(169,179)(170,178)(171,177)(172,176)(173,175)(181,187)(182,186)(183,185)(188,192)(189,191)(193,199)(194,198)(195,197)(200,204)(201,203)(206,216)(207,215)(208,214)(209,213)(210,212)(217,219)(220,228)(221,227)(222,226)(223,225)(229,235)(230,234)(231,233)(236,240)(237,239), (1,130,40,183,111)(2,131,41,184,112)(3,132,42,185,113)(4,121,43,186,114)(5,122,44,187,115)(6,123,45,188,116)(7,124,46,189,117)(8,125,47,190,118)(9,126,48,191,119)(10,127,37,192,120)(11,128,38,181,109)(12,129,39,182,110)(13,164,215,236,67)(14,165,216,237,68)(15,166,205,238,69)(16,167,206,239,70)(17,168,207,240,71)(18,157,208,229,72)(19,158,209,230,61)(20,159,210,231,62)(21,160,211,232,63)(22,161,212,233,64)(23,162,213,234,65)(24,163,214,235,66)(25,173,217,82,95)(26,174,218,83,96)(27,175,219,84,85)(28,176,220,73,86)(29,177,221,74,87)(30,178,222,75,88)(31,179,223,76,89)(32,180,224,77,90)(33,169,225,78,91)(34,170,226,79,92)(35,171,227,80,93)(36,172,228,81,94)(49,197,105,139,152)(50,198,106,140,153)(51,199,107,141,154)(52,200,108,142,155)(53,201,97,143,156)(54,202,98,144,145)(55,203,99,133,146)(56,204,100,134,147)(57,193,101,135,148)(58,194,102,136,149)(59,195,103,137,150)(60,196,104,138,151), (1,25,143,165,7,31,137,159)(2,26,144,166,8,32,138,160)(3,27,133,167,9,33,139,161)(4,28,134,168,10,34,140,162)(5,29,135,157,11,35,141,163)(6,30,136,158,12,36,142,164)(13,188,178,58,19,182,172,52)(14,189,179,59,20,183,173,53)(15,190,180,60,21,184,174,54)(16,191,169,49,22,185,175,55)(17,192,170,50,23,186,176,56)(18,181,171,51,24,187,177,57)(37,92,198,213,43,86,204,207)(38,93,199,214,44,87,193,208)(39,94,200,215,45,88,194,209)(40,95,201,216,46,89,195,210)(41,96,202,205,47,90,196,211)(42,85,203,206,48,91,197,212)(61,129,228,108,67,123,222,102)(62,130,217,97,68,124,223,103)(63,131,218,98,69,125,224,104)(64,132,219,99,70,126,225,105)(65,121,220,100,71,127,226,106)(66,122,221,101,72,128,227,107)(73,147,240,120,79,153,234,114)(74,148,229,109,80,154,235,115)(75,149,230,110,81,155,236,116)(76,150,231,111,82,156,237,117)(77,151,232,112,83,145,238,118)(78,152,233,113,84,146,239,119) );
G=PermutationGroup([[(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,27),(28,36),(29,35),(30,34),(31,33),(37,45),(38,44),(39,43),(40,42),(46,48),(49,59),(50,58),(51,57),(52,56),(53,55),(61,65),(62,64),(66,72),(67,71),(68,70),(73,81),(74,80),(75,79),(76,78),(82,84),(85,95),(86,94),(87,93),(88,92),(89,91),(97,99),(100,108),(101,107),(102,106),(103,105),(109,115),(110,114),(111,113),(116,120),(117,119),(121,129),(122,128),(123,127),(124,126),(130,132),(133,143),(134,142),(135,141),(136,140),(137,139),(146,156),(147,155),(148,154),(149,153),(150,152),(157,163),(158,162),(159,161),(164,168),(165,167),(169,179),(170,178),(171,177),(172,176),(173,175),(181,187),(182,186),(183,185),(188,192),(189,191),(193,199),(194,198),(195,197),(200,204),(201,203),(206,216),(207,215),(208,214),(209,213),(210,212),(217,219),(220,228),(221,227),(222,226),(223,225),(229,235),(230,234),(231,233),(236,240),(237,239)], [(1,130,40,183,111),(2,131,41,184,112),(3,132,42,185,113),(4,121,43,186,114),(5,122,44,187,115),(6,123,45,188,116),(7,124,46,189,117),(8,125,47,190,118),(9,126,48,191,119),(10,127,37,192,120),(11,128,38,181,109),(12,129,39,182,110),(13,164,215,236,67),(14,165,216,237,68),(15,166,205,238,69),(16,167,206,239,70),(17,168,207,240,71),(18,157,208,229,72),(19,158,209,230,61),(20,159,210,231,62),(21,160,211,232,63),(22,161,212,233,64),(23,162,213,234,65),(24,163,214,235,66),(25,173,217,82,95),(26,174,218,83,96),(27,175,219,84,85),(28,176,220,73,86),(29,177,221,74,87),(30,178,222,75,88),(31,179,223,76,89),(32,180,224,77,90),(33,169,225,78,91),(34,170,226,79,92),(35,171,227,80,93),(36,172,228,81,94),(49,197,105,139,152),(50,198,106,140,153),(51,199,107,141,154),(52,200,108,142,155),(53,201,97,143,156),(54,202,98,144,145),(55,203,99,133,146),(56,204,100,134,147),(57,193,101,135,148),(58,194,102,136,149),(59,195,103,137,150),(60,196,104,138,151)], [(1,25,143,165,7,31,137,159),(2,26,144,166,8,32,138,160),(3,27,133,167,9,33,139,161),(4,28,134,168,10,34,140,162),(5,29,135,157,11,35,141,163),(6,30,136,158,12,36,142,164),(13,188,178,58,19,182,172,52),(14,189,179,59,20,183,173,53),(15,190,180,60,21,184,174,54),(16,191,169,49,22,185,175,55),(17,192,170,50,23,186,176,56),(18,181,171,51,24,187,177,57),(37,92,198,213,43,86,204,207),(38,93,199,214,44,87,193,208),(39,94,200,215,45,88,194,209),(40,95,201,216,46,89,195,210),(41,96,202,205,47,90,196,211),(42,85,203,206,48,91,197,212),(61,129,228,108,67,123,222,102),(62,130,217,97,68,124,223,103),(63,131,218,98,69,125,224,104),(64,132,219,99,70,126,225,105),(65,121,220,100,71,127,226,106),(66,122,221,101,72,128,227,107),(73,147,240,120,79,153,234,114),(74,148,229,109,80,154,235,115),(75,149,230,110,81,155,236,116),(76,150,231,111,82,156,237,117),(77,151,232,112,83,145,238,118),(78,152,233,113,84,146,239,119)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20 | 24A | ··· | 24H | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 24 | ··· | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 6 | 6 | 10 | 2 | 2 | 5 | 5 | 30 | 30 | 4 | 2 | 10 | 10 | 5 | 5 | 5 | 5 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 24 | 24 | 2 | 2 | 10 | 10 | 8 | 8 | 10 | ··· | 10 | 8 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | F5 | C2×F5 | C2×F5 | S3×F5 | D4.F5 | C2×S3×F5 | D12.2F5 |
| kernel | D12.2F5 | S3×C5⋊C8 | D6.F5 | C3×D5⋊C8 | C12.F5 | D12⋊5D5 | C15⋊D4 | C5×D12 | Dic30 | D5⋊C8 | C5⋊C8 | C4×D5 | C20 | D10 | C15 | C5 | D12 | C12 | D6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D12.2F5 ►in GL8(𝔽241)
| 56 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 79 | 185 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 198 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 188 | 77 | 141 |
| 0 | 0 | 0 | 0 | 95 | 88 | 135 | 159 |
| 0 | 0 | 0 | 0 | 153 | 106 | 82 | 236 |
| 0 | 0 | 0 | 0 | 100 | 183 | 223 | 53 |
G:=sub<GL(8,GF(241))| [56,79,0,0,0,0,0,0,3,185,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[240,198,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[211,0,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,18,95,153,100,0,0,0,0,188,88,106,183,0,0,0,0,77,135,82,223,0,0,0,0,141,159,236,53] >;
D12.2F5 in GAP, Magma, Sage, TeX
D_{12}._2F_5 % in TeX
G:=Group("D12.2F5"); // GroupNames label
G:=SmallGroup(480,987);
// by ID
G=gap.SmallGroup(480,987);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,100,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=1,d^4=a^6,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^3>;
// generators/relations