metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.F5, Dic30.1C4, C5⋊C8.1D6, C4.F5⋊3S3, C15⋊2(C8○D4), C60.6(C2×C4), D6.F5⋊2C2, C3⋊2(D4.F5), D6.2(C2×F5), C4.13(S3×F5), C20.12(C4×S3), C5⋊2(D12.C4), (C4×D5).34D6, D10.2(C4×S3), (C5×D12).1C4, C15⋊D4.2C4, C12.16(C2×F5), C60.C4⋊2C2, C6.8(C22×F5), C30.8(C22×C4), D12⋊5D5.4C2, C15⋊C8.4C22, Dic15.4(C2×C4), (D5×C12).35C22, (S3×Dic5).6C22, (C3×Dic5).26C23, Dic5.28(C22×S3), (S3×C5⋊C8)⋊3C2, C10.8(S3×C2×C4), C2.12(C2×S3×F5), (C3×C4.F5)⋊3C2, (C3×C5⋊C8).1C22, (S3×C10).2(C2×C4), (C6×D5).17(C2×C4), SmallGroup(480,989)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.F5
G = < a,b,c,d | a12=b2=c5=1, d4=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c3 >
Subgroups: 548 in 124 conjugacy classes, 46 normal (28 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, C2×C3⋊C8, C3×M4(2), C4○D12, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, D4⋊2D5, D12.C4, C3×C5⋊C8, C15⋊C8, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, D4.F5, S3×C5⋊C8, D6.F5, C3×C4.F5, C60.C4, D12⋊5D5, D12.F5
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, D12.C4, S3×F5, D4.F5, C2×S3×F5, D12.F5
►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)(14 24)(15 23)(16 22)(17 21)(18 20)(26 36)(27 35)(28 34)(29 33)(30 32)(37 43)(38 42)(39 41)(44 48)(45 47)(49 59)(50 58)(51 57)(52 56)(53 55)(62 72)(63 71)(64 70)(65 69)(66 68)(73 83)(74 82)(75 81)(76 80)(77 79)(85 87)(88 96)(89 95)(90 94)(91 93)(97 107)(98 106)(99 105)(100 104)(101 103)(109 111)(112 120)(113 119)(114 118)(115 117)(121 129)(122 128)(123 127)(124 126)(130 132)(133 143)(134 142)(135 141)(136 140)(137 139)(145 155)(146 154)(147 153)(148 152)(149 151)(158 168)(159 167)(160 166)(161 165)(162 164)(169 173)(170 172)(174 180)(175 179)(176 178)(181 183)(184 192)(185 191)(186 190)(187 189)(193 199)(194 198)(195 197)(200 204)(201 203)(205 215)(206 214)(207 213)(208 212)(209 211)(217 219)(220 228)(221 227)(222 226)(223 225)(229 233)(230 232)(234 240)(235 239)(236 238)
(1 107 30 217 24)(2 108 31 218 13)(3 97 32 219 14)(4 98 33 220 15)(5 99 34 221 16)(6 100 35 222 17)(7 101 36 223 18)(8 102 25 224 19)(9 103 26 225 20)(10 104 27 226 21)(11 105 28 227 22)(12 106 29 228 23)(37 89 213 234 174)(38 90 214 235 175)(39 91 215 236 176)(40 92 216 237 177)(41 93 205 238 178)(42 94 206 239 179)(43 95 207 240 180)(44 96 208 229 169)(45 85 209 230 170)(46 86 210 231 171)(47 87 211 232 172)(48 88 212 233 173)(49 189 73 197 139)(50 190 74 198 140)(51 191 75 199 141)(52 192 76 200 142)(53 181 77 201 143)(54 182 78 202 144)(55 183 79 203 133)(56 184 80 204 134)(57 185 81 193 135)(58 186 82 194 136)(59 187 83 195 137)(60 188 84 196 138)(61 131 163 116 150)(62 132 164 117 151)(63 121 165 118 152)(64 122 166 119 153)(65 123 167 120 154)(66 124 168 109 155)(67 125 157 110 156)(68 126 158 111 145)(69 127 159 112 146)(70 128 160 113 147)(71 129 161 114 148)(72 130 162 115 149)
(1 140 170 127 7 134 176 121)(2 135 171 122 8 141 177 128)(3 142 172 129 9 136 178 123)(4 137 173 124 10 143 179 130)(5 144 174 131 11 138 180 125)(6 139 175 126 12 133 169 132)(13 81 46 119 19 75 40 113)(14 76 47 114 20 82 41 120)(15 83 48 109 21 77 42 115)(16 78 37 116 22 84 43 110)(17 73 38 111 23 79 44 117)(18 80 39 118 24 74 45 112)(25 199 216 160 31 193 210 166)(26 194 205 167 32 200 211 161)(27 201 206 162 33 195 212 168)(28 196 207 157 34 202 213 163)(29 203 208 164 35 197 214 158)(30 198 209 159 36 204 215 165)(49 90 68 228 55 96 62 222)(50 85 69 223 56 91 63 217)(51 92 70 218 57 86 64 224)(52 87 71 225 58 93 65 219)(53 94 72 220 59 88 66 226)(54 89 61 227 60 95 67 221)(97 192 232 148 103 186 238 154)(98 187 233 155 104 181 239 149)(99 182 234 150 105 188 240 156)(100 189 235 145 106 183 229 151)(101 184 236 152 107 190 230 146)(102 191 237 147 108 185 231 153)
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)(14,24)(15,23)(16,22)(17,21)(18,20)(26,36)(27,35)(28,34)(29,33)(30,32)(37,43)(38,42)(39,41)(44,48)(45,47)(49,59)(50,58)(51,57)(52,56)(53,55)(62,72)(63,71)(64,70)(65,69)(66,68)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93)(97,107)(98,106)(99,105)(100,104)(101,103)(109,111)(112,120)(113,119)(114,118)(115,117)(121,129)(122,128)(123,127)(124,126)(130,132)(133,143)(134,142)(135,141)(136,140)(137,139)(145,155)(146,154)(147,153)(148,152)(149,151)(158,168)(159,167)(160,166)(161,165)(162,164)(169,173)(170,172)(174,180)(175,179)(176,178)(181,183)(184,192)(185,191)(186,190)(187,189)(193,199)(194,198)(195,197)(200,204)(201,203)(205,215)(206,214)(207,213)(208,212)(209,211)(217,219)(220,228)(221,227)(222,226)(223,225)(229,233)(230,232)(234,240)(235,239)(236,238), (1,107,30,217,24)(2,108,31,218,13)(3,97,32,219,14)(4,98,33,220,15)(5,99,34,221,16)(6,100,35,222,17)(7,101,36,223,18)(8,102,25,224,19)(9,103,26,225,20)(10,104,27,226,21)(11,105,28,227,22)(12,106,29,228,23)(37,89,213,234,174)(38,90,214,235,175)(39,91,215,236,176)(40,92,216,237,177)(41,93,205,238,178)(42,94,206,239,179)(43,95,207,240,180)(44,96,208,229,169)(45,85,209,230,170)(46,86,210,231,171)(47,87,211,232,172)(48,88,212,233,173)(49,189,73,197,139)(50,190,74,198,140)(51,191,75,199,141)(52,192,76,200,142)(53,181,77,201,143)(54,182,78,202,144)(55,183,79,203,133)(56,184,80,204,134)(57,185,81,193,135)(58,186,82,194,136)(59,187,83,195,137)(60,188,84,196,138)(61,131,163,116,150)(62,132,164,117,151)(63,121,165,118,152)(64,122,166,119,153)(65,123,167,120,154)(66,124,168,109,155)(67,125,157,110,156)(68,126,158,111,145)(69,127,159,112,146)(70,128,160,113,147)(71,129,161,114,148)(72,130,162,115,149), (1,140,170,127,7,134,176,121)(2,135,171,122,8,141,177,128)(3,142,172,129,9,136,178,123)(4,137,173,124,10,143,179,130)(5,144,174,131,11,138,180,125)(6,139,175,126,12,133,169,132)(13,81,46,119,19,75,40,113)(14,76,47,114,20,82,41,120)(15,83,48,109,21,77,42,115)(16,78,37,116,22,84,43,110)(17,73,38,111,23,79,44,117)(18,80,39,118,24,74,45,112)(25,199,216,160,31,193,210,166)(26,194,205,167,32,200,211,161)(27,201,206,162,33,195,212,168)(28,196,207,157,34,202,213,163)(29,203,208,164,35,197,214,158)(30,198,209,159,36,204,215,165)(49,90,68,228,55,96,62,222)(50,85,69,223,56,91,63,217)(51,92,70,218,57,86,64,224)(52,87,71,225,58,93,65,219)(53,94,72,220,59,88,66,226)(54,89,61,227,60,95,67,221)(97,192,232,148,103,186,238,154)(98,187,233,155,104,181,239,149)(99,182,234,150,105,188,240,156)(100,189,235,145,106,183,229,151)(101,184,236,152,107,190,230,146)(102,191,237,147,108,185,231,153)>;
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)(14,24)(15,23)(16,22)(17,21)(18,20)(26,36)(27,35)(28,34)(29,33)(30,32)(37,43)(38,42)(39,41)(44,48)(45,47)(49,59)(50,58)(51,57)(52,56)(53,55)(62,72)(63,71)(64,70)(65,69)(66,68)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93)(97,107)(98,106)(99,105)(100,104)(101,103)(109,111)(112,120)(113,119)(114,118)(115,117)(121,129)(122,128)(123,127)(124,126)(130,132)(133,143)(134,142)(135,141)(136,140)(137,139)(145,155)(146,154)(147,153)(148,152)(149,151)(158,168)(159,167)(160,166)(161,165)(162,164)(169,173)(170,172)(174,180)(175,179)(176,178)(181,183)(184,192)(185,191)(186,190)(187,189)(193,199)(194,198)(195,197)(200,204)(201,203)(205,215)(206,214)(207,213)(208,212)(209,211)(217,219)(220,228)(221,227)(222,226)(223,225)(229,233)(230,232)(234,240)(235,239)(236,238), (1,107,30,217,24)(2,108,31,218,13)(3,97,32,219,14)(4,98,33,220,15)(5,99,34,221,16)(6,100,35,222,17)(7,101,36,223,18)(8,102,25,224,19)(9,103,26,225,20)(10,104,27,226,21)(11,105,28,227,22)(12,106,29,228,23)(37,89,213,234,174)(38,90,214,235,175)(39,91,215,236,176)(40,92,216,237,177)(41,93,205,238,178)(42,94,206,239,179)(43,95,207,240,180)(44,96,208,229,169)(45,85,209,230,170)(46,86,210,231,171)(47,87,211,232,172)(48,88,212,233,173)(49,189,73,197,139)(50,190,74,198,140)(51,191,75,199,141)(52,192,76,200,142)(53,181,77,201,143)(54,182,78,202,144)(55,183,79,203,133)(56,184,80,204,134)(57,185,81,193,135)(58,186,82,194,136)(59,187,83,195,137)(60,188,84,196,138)(61,131,163,116,150)(62,132,164,117,151)(63,121,165,118,152)(64,122,166,119,153)(65,123,167,120,154)(66,124,168,109,155)(67,125,157,110,156)(68,126,158,111,145)(69,127,159,112,146)(70,128,160,113,147)(71,129,161,114,148)(72,130,162,115,149), (1,140,170,127,7,134,176,121)(2,135,171,122,8,141,177,128)(3,142,172,129,9,136,178,123)(4,137,173,124,10,143,179,130)(5,144,174,131,11,138,180,125)(6,139,175,126,12,133,169,132)(13,81,46,119,19,75,40,113)(14,76,47,114,20,82,41,120)(15,83,48,109,21,77,42,115)(16,78,37,116,22,84,43,110)(17,73,38,111,23,79,44,117)(18,80,39,118,24,74,45,112)(25,199,216,160,31,193,210,166)(26,194,205,167,32,200,211,161)(27,201,206,162,33,195,212,168)(28,196,207,157,34,202,213,163)(29,203,208,164,35,197,214,158)(30,198,209,159,36,204,215,165)(49,90,68,228,55,96,62,222)(50,85,69,223,56,91,63,217)(51,92,70,218,57,86,64,224)(52,87,71,225,58,93,65,219)(53,94,72,220,59,88,66,226)(54,89,61,227,60,95,67,221)(97,192,232,148,103,186,238,154)(98,187,233,155,104,181,239,149)(99,182,234,150,105,188,240,156)(100,189,235,145,106,183,229,151)(101,184,236,152,107,190,230,146)(102,191,237,147,108,185,231,153) );
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),(14,24),(15,23),(16,22),(17,21),(18,20),(26,36),(27,35),(28,34),(29,33),(30,32),(37,43),(38,42),(39,41),(44,48),(45,47),(49,59),(50,58),(51,57),(52,56),(53,55),(62,72),(63,71),(64,70),(65,69),(66,68),(73,83),(74,82),(75,81),(76,80),(77,79),(85,87),(88,96),(89,95),(90,94),(91,93),(97,107),(98,106),(99,105),(100,104),(101,103),(109,111),(112,120),(113,119),(114,118),(115,117),(121,129),(122,128),(123,127),(124,126),(130,132),(133,143),(134,142),(135,141),(136,140),(137,139),(145,155),(146,154),(147,153),(148,152),(149,151),(158,168),(159,167),(160,166),(161,165),(162,164),(169,173),(170,172),(174,180),(175,179),(176,178),(181,183),(184,192),(185,191),(186,190),(187,189),(193,199),(194,198),(195,197),(200,204),(201,203),(205,215),(206,214),(207,213),(208,212),(209,211),(217,219),(220,228),(221,227),(222,226),(223,225),(229,233),(230,232),(234,240),(235,239),(236,238)], [(1,107,30,217,24),(2,108,31,218,13),(3,97,32,219,14),(4,98,33,220,15),(5,99,34,221,16),(6,100,35,222,17),(7,101,36,223,18),(8,102,25,224,19),(9,103,26,225,20),(10,104,27,226,21),(11,105,28,227,22),(12,106,29,228,23),(37,89,213,234,174),(38,90,214,235,175),(39,91,215,236,176),(40,92,216,237,177),(41,93,205,238,178),(42,94,206,239,179),(43,95,207,240,180),(44,96,208,229,169),(45,85,209,230,170),(46,86,210,231,171),(47,87,211,232,172),(48,88,212,233,173),(49,189,73,197,139),(50,190,74,198,140),(51,191,75,199,141),(52,192,76,200,142),(53,181,77,201,143),(54,182,78,202,144),(55,183,79,203,133),(56,184,80,204,134),(57,185,81,193,135),(58,186,82,194,136),(59,187,83,195,137),(60,188,84,196,138),(61,131,163,116,150),(62,132,164,117,151),(63,121,165,118,152),(64,122,166,119,153),(65,123,167,120,154),(66,124,168,109,155),(67,125,157,110,156),(68,126,158,111,145),(69,127,159,112,146),(70,128,160,113,147),(71,129,161,114,148),(72,130,162,115,149)], [(1,140,170,127,7,134,176,121),(2,135,171,122,8,141,177,128),(3,142,172,129,9,136,178,123),(4,137,173,124,10,143,179,130),(5,144,174,131,11,138,180,125),(6,139,175,126,12,133,169,132),(13,81,46,119,19,75,40,113),(14,76,47,114,20,82,41,120),(15,83,48,109,21,77,42,115),(16,78,37,116,22,84,43,110),(17,73,38,111,23,79,44,117),(18,80,39,118,24,74,45,112),(25,199,216,160,31,193,210,166),(26,194,205,167,32,200,211,161),(27,201,206,162,33,195,212,168),(28,196,207,157,34,202,213,163),(29,203,208,164,35,197,214,158),(30,198,209,159,36,204,215,165),(49,90,68,228,55,96,62,222),(50,85,69,223,56,91,63,217),(51,92,70,218,57,86,64,224),(52,87,71,225,58,93,65,219),(53,94,72,220,59,88,66,226),(54,89,61,227,60,95,67,221),(97,192,232,148,103,186,238,154),(98,187,233,155,104,181,239,149),(99,182,234,150,105,188,240,156),(100,189,235,145,106,183,229,151),(101,184,236,152,107,190,230,146),(102,191,237,147,108,185,231,153)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20 | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 6 | 6 | 10 | 2 | 2 | 5 | 5 | 30 | 30 | 4 | 2 | 20 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 4 | 24 | 24 | 4 | 10 | 10 | 8 | 8 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | F5 | C2×F5 | C2×F5 | D12.C4 | S3×F5 | D4.F5 | C2×S3×F5 | D12.F5 |
| kernel | D12.F5 | S3×C5⋊C8 | D6.F5 | C3×C4.F5 | C60.C4 | D12⋊5D5 | C15⋊D4 | C5×D12 | Dic30 | C4.F5 | C5⋊C8 | C4×D5 | C20 | D10 | C15 | D12 | C12 | D6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D12.F5 ►in GL8(𝔽241)
| 240 | 240 | 81 | 79 | 0 | 0 | 0 | 0 |
| 1 | 0 | 81 | 81 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 2 | 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 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 239 | 1 | 240 | 0 | 0 | 0 | 0 |
| 239 | 240 | 0 | 240 | 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 | 1 |
| 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 |
| 98 | 5 | 144 | 144 | 0 | 0 | 0 | 0 |
| 236 | 93 | 194 | 144 | 0 | 0 | 0 | 0 |
| 113 | 113 | 148 | 236 | 0 | 0 | 0 | 0 |
| 0 | 113 | 5 | 143 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 120 | 21 | 224 | 227 |
| 0 | 0 | 0 | 0 | 103 | 7 | 114 | 106 |
| 0 | 0 | 0 | 0 | 234 | 127 | 135 | 89 |
| 0 | 0 | 0 | 0 | 14 | 110 | 121 | 220 |
G:=sub<GL(8,GF(241))| [240,1,2,1,0,0,0,0,240,0,1,2,0,0,0,0,81,81,0,240,0,0,0,0,79,81,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,240,239,0,0,0,0,1,240,239,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[98,236,113,0,0,0,0,0,5,93,113,113,0,0,0,0,144,194,148,5,0,0,0,0,144,144,236,143,0,0,0,0,0,0,0,0,120,103,234,14,0,0,0,0,21,7,127,110,0,0,0,0,224,114,135,121,0,0,0,0,227,106,89,220] >;
D12.F5 in GAP, Magma, Sage, TeX
D_{12}.F_5 % in TeX
G:=Group("D12.F5"); // GroupNames label
G:=SmallGroup(480,989);
// by ID
G=gap.SmallGroup(480,989);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,100,80,1356,9414,2379]);
// 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,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;
// generators/relations