metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.27D10, C60.37C23, Dic10.14D6, Dic30.13C22, C3⋊C8.9D10, (Q8×D5)⋊5S3, Q8⋊2S3⋊4D5, (C6×D5).66D4, C3⋊Dic20⋊7C2, C15⋊7Q16⋊5C2, (C4×D5).11D6, C6.150(D4×D5), Q8.16(S3×D5), (C5×Q8).22D6, C20.D6⋊7C2, C30.199(C2×D4), C3⋊7(SD16⋊D5), (C3×Q8).20D10, D12⋊5D5.1C2, C20.32D6⋊7C2, C5⋊2(Q8.11D6), C15⋊19(C8.C22), C20.37(C22×S3), (C3×Dic5).17D4, C12.37(C22×D5), (Q8×C15).7C22, D10.30(C3⋊D4), C15⋊3C8.11C22, (D5×C12).13C22, (C5×D12).13C22, Dic5.24(C3⋊D4), (C3×Dic10).13C22, (C3×Q8×D5)⋊2C2, C4.37(C2×S3×D5), C2.32(D5×C3⋊D4), (C5×Q8⋊2S3)⋊5C2, C10.53(C2×C3⋊D4), (C5×C3⋊C8).11C22, SmallGroup(480,589)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.27D10
G = < a,b,c,d | a12=b2=1, c10=d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=c9 >
Subgroups: 588 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C5×S3, C3×D5, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4.Dic3, Q8⋊2S3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C3×Dic5, C3×Dic5, Dic15, C60, C60, C6×D5, S3×C10, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, Q8.11D6, C5×C3⋊C8, C15⋊3C8, S3×Dic5, C15⋊D4, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×D12, Dic30, Q8×C15, SD16⋊D5, C20.32D6, C20.D6, C3⋊Dic20, C5×Q8⋊2S3, C15⋊7Q16, D12⋊5D5, C3×Q8×D5, D12.27D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.11D6, C2×S3×D5, SD16⋊D5, D5×C3⋊D4, D12.27D10
►On 240 points
Generators in S240
(1 24 171 193 221 51 11 34 161 183 231 41)(2 35 172 184 222 42 12 25 162 194 232 52)(3 26 173 195 223 53 13 36 163 185 233 43)(4 37 174 186 224 44 14 27 164 196 234 54)(5 28 175 197 225 55 15 38 165 187 235 45)(6 39 176 188 226 46 16 29 166 198 236 56)(7 30 177 199 227 57 17 40 167 189 237 47)(8 21 178 190 228 48 18 31 168 200 238 58)(9 32 179 181 229 59 19 22 169 191 239 49)(10 23 180 192 230 50 20 33 170 182 240 60)(61 104 215 152 134 90 71 114 205 142 124 100)(62 115 216 143 135 81 72 105 206 153 125 91)(63 106 217 154 136 92 73 116 207 144 126 82)(64 117 218 145 137 83 74 107 208 155 127 93)(65 108 219 156 138 94 75 118 209 146 128 84)(66 119 220 147 139 85 76 109 210 157 129 95)(67 110 201 158 140 96 77 120 211 148 130 86)(68 101 202 149 121 87 78 111 212 159 131 97)(69 112 203 160 122 98 79 102 213 150 132 88)(70 103 204 151 123 89 80 113 214 141 133 99)
(1 123)(2 104)(3 125)(4 106)(5 127)(6 108)(7 129)(8 110)(9 131)(10 112)(11 133)(12 114)(13 135)(14 116)(15 137)(16 118)(17 139)(18 120)(19 121)(20 102)(21 67)(22 149)(23 69)(24 151)(25 71)(26 153)(27 73)(28 155)(29 75)(30 157)(31 77)(32 159)(33 79)(34 141)(35 61)(36 143)(37 63)(38 145)(39 65)(40 147)(41 89)(42 205)(43 91)(44 207)(45 93)(46 209)(47 95)(48 211)(49 97)(50 213)(51 99)(52 215)(53 81)(54 217)(55 83)(56 219)(57 85)(58 201)(59 87)(60 203)(62 233)(64 235)(66 237)(68 239)(70 221)(72 223)(74 225)(76 227)(78 229)(80 231)(82 174)(84 176)(86 178)(88 180)(90 162)(92 164)(94 166)(96 168)(98 170)(100 172)(101 191)(103 193)(105 195)(107 197)(109 199)(111 181)(113 183)(115 185)(117 187)(119 189)(122 182)(124 184)(126 186)(128 188)(130 190)(132 192)(134 194)(136 196)(138 198)(140 200)(142 222)(144 224)(146 226)(148 228)(150 230)(152 232)(154 234)(156 236)(158 238)(160 240)(161 214)(163 216)(165 218)(167 220)(169 202)(171 204)(173 206)(175 208)(177 210)(179 212)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 50 51 60)(42 59 52 49)(43 48 53 58)(44 57 54 47)(45 46 55 56)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)(81 96 91 86)(82 85 92 95)(83 94 93 84)(87 90 97 100)(88 99 98 89)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 124 131 134)(122 133 132 123)(125 140 135 130)(126 129 136 139)(127 138 137 128)(141 160 151 150)(142 149 152 159)(143 158 153 148)(144 147 154 157)(145 156 155 146)(161 170 171 180)(162 179 172 169)(163 168 173 178)(164 177 174 167)(165 166 175 176)(181 194 191 184)(182 183 192 193)(185 190 195 200)(186 199 196 189)(187 188 197 198)(201 216 211 206)(202 205 212 215)(203 214 213 204)(207 210 217 220)(208 219 218 209)(221 230 231 240)(222 239 232 229)(223 228 233 238)(224 237 234 227)(225 226 235 236)
G:=sub<Sym(240)| (1,24,171,193,221,51,11,34,161,183,231,41)(2,35,172,184,222,42,12,25,162,194,232,52)(3,26,173,195,223,53,13,36,163,185,233,43)(4,37,174,186,224,44,14,27,164,196,234,54)(5,28,175,197,225,55,15,38,165,187,235,45)(6,39,176,188,226,46,16,29,166,198,236,56)(7,30,177,199,227,57,17,40,167,189,237,47)(8,21,178,190,228,48,18,31,168,200,238,58)(9,32,179,181,229,59,19,22,169,191,239,49)(10,23,180,192,230,50,20,33,170,182,240,60)(61,104,215,152,134,90,71,114,205,142,124,100)(62,115,216,143,135,81,72,105,206,153,125,91)(63,106,217,154,136,92,73,116,207,144,126,82)(64,117,218,145,137,83,74,107,208,155,127,93)(65,108,219,156,138,94,75,118,209,146,128,84)(66,119,220,147,139,85,76,109,210,157,129,95)(67,110,201,158,140,96,77,120,211,148,130,86)(68,101,202,149,121,87,78,111,212,159,131,97)(69,112,203,160,122,98,79,102,213,150,132,88)(70,103,204,151,123,89,80,113,214,141,133,99), (1,123)(2,104)(3,125)(4,106)(5,127)(6,108)(7,129)(8,110)(9,131)(10,112)(11,133)(12,114)(13,135)(14,116)(15,137)(16,118)(17,139)(18,120)(19,121)(20,102)(21,67)(22,149)(23,69)(24,151)(25,71)(26,153)(27,73)(28,155)(29,75)(30,157)(31,77)(32,159)(33,79)(34,141)(35,61)(36,143)(37,63)(38,145)(39,65)(40,147)(41,89)(42,205)(43,91)(44,207)(45,93)(46,209)(47,95)(48,211)(49,97)(50,213)(51,99)(52,215)(53,81)(54,217)(55,83)(56,219)(57,85)(58,201)(59,87)(60,203)(62,233)(64,235)(66,237)(68,239)(70,221)(72,223)(74,225)(76,227)(78,229)(80,231)(82,174)(84,176)(86,178)(88,180)(90,162)(92,164)(94,166)(96,168)(98,170)(100,172)(101,191)(103,193)(105,195)(107,197)(109,199)(111,181)(113,183)(115,185)(117,187)(119,189)(122,182)(124,184)(126,186)(128,188)(130,190)(132,192)(134,194)(136,196)(138,198)(140,200)(142,222)(144,224)(146,226)(148,228)(150,230)(152,232)(154,234)(156,236)(158,238)(160,240)(161,214)(163,216)(165,218)(167,220)(169,202)(171,204)(173,206)(175,208)(177,210)(179,212), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,124,131,134)(122,133,132,123)(125,140,135,130)(126,129,136,139)(127,138,137,128)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146)(161,170,171,180)(162,179,172,169)(163,168,173,178)(164,177,174,167)(165,166,175,176)(181,194,191,184)(182,183,192,193)(185,190,195,200)(186,199,196,189)(187,188,197,198)(201,216,211,206)(202,205,212,215)(203,214,213,204)(207,210,217,220)(208,219,218,209)(221,230,231,240)(222,239,232,229)(223,228,233,238)(224,237,234,227)(225,226,235,236)>;
G:=Group( (1,24,171,193,221,51,11,34,161,183,231,41)(2,35,172,184,222,42,12,25,162,194,232,52)(3,26,173,195,223,53,13,36,163,185,233,43)(4,37,174,186,224,44,14,27,164,196,234,54)(5,28,175,197,225,55,15,38,165,187,235,45)(6,39,176,188,226,46,16,29,166,198,236,56)(7,30,177,199,227,57,17,40,167,189,237,47)(8,21,178,190,228,48,18,31,168,200,238,58)(9,32,179,181,229,59,19,22,169,191,239,49)(10,23,180,192,230,50,20,33,170,182,240,60)(61,104,215,152,134,90,71,114,205,142,124,100)(62,115,216,143,135,81,72,105,206,153,125,91)(63,106,217,154,136,92,73,116,207,144,126,82)(64,117,218,145,137,83,74,107,208,155,127,93)(65,108,219,156,138,94,75,118,209,146,128,84)(66,119,220,147,139,85,76,109,210,157,129,95)(67,110,201,158,140,96,77,120,211,148,130,86)(68,101,202,149,121,87,78,111,212,159,131,97)(69,112,203,160,122,98,79,102,213,150,132,88)(70,103,204,151,123,89,80,113,214,141,133,99), (1,123)(2,104)(3,125)(4,106)(5,127)(6,108)(7,129)(8,110)(9,131)(10,112)(11,133)(12,114)(13,135)(14,116)(15,137)(16,118)(17,139)(18,120)(19,121)(20,102)(21,67)(22,149)(23,69)(24,151)(25,71)(26,153)(27,73)(28,155)(29,75)(30,157)(31,77)(32,159)(33,79)(34,141)(35,61)(36,143)(37,63)(38,145)(39,65)(40,147)(41,89)(42,205)(43,91)(44,207)(45,93)(46,209)(47,95)(48,211)(49,97)(50,213)(51,99)(52,215)(53,81)(54,217)(55,83)(56,219)(57,85)(58,201)(59,87)(60,203)(62,233)(64,235)(66,237)(68,239)(70,221)(72,223)(74,225)(76,227)(78,229)(80,231)(82,174)(84,176)(86,178)(88,180)(90,162)(92,164)(94,166)(96,168)(98,170)(100,172)(101,191)(103,193)(105,195)(107,197)(109,199)(111,181)(113,183)(115,185)(117,187)(119,189)(122,182)(124,184)(126,186)(128,188)(130,190)(132,192)(134,194)(136,196)(138,198)(140,200)(142,222)(144,224)(146,226)(148,228)(150,230)(152,232)(154,234)(156,236)(158,238)(160,240)(161,214)(163,216)(165,218)(167,220)(169,202)(171,204)(173,206)(175,208)(177,210)(179,212), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,124,131,134)(122,133,132,123)(125,140,135,130)(126,129,136,139)(127,138,137,128)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146)(161,170,171,180)(162,179,172,169)(163,168,173,178)(164,177,174,167)(165,166,175,176)(181,194,191,184)(182,183,192,193)(185,190,195,200)(186,199,196,189)(187,188,197,198)(201,216,211,206)(202,205,212,215)(203,214,213,204)(207,210,217,220)(208,219,218,209)(221,230,231,240)(222,239,232,229)(223,228,233,238)(224,237,234,227)(225,226,235,236) );
G=PermutationGroup([[(1,24,171,193,221,51,11,34,161,183,231,41),(2,35,172,184,222,42,12,25,162,194,232,52),(3,26,173,195,223,53,13,36,163,185,233,43),(4,37,174,186,224,44,14,27,164,196,234,54),(5,28,175,197,225,55,15,38,165,187,235,45),(6,39,176,188,226,46,16,29,166,198,236,56),(7,30,177,199,227,57,17,40,167,189,237,47),(8,21,178,190,228,48,18,31,168,200,238,58),(9,32,179,181,229,59,19,22,169,191,239,49),(10,23,180,192,230,50,20,33,170,182,240,60),(61,104,215,152,134,90,71,114,205,142,124,100),(62,115,216,143,135,81,72,105,206,153,125,91),(63,106,217,154,136,92,73,116,207,144,126,82),(64,117,218,145,137,83,74,107,208,155,127,93),(65,108,219,156,138,94,75,118,209,146,128,84),(66,119,220,147,139,85,76,109,210,157,129,95),(67,110,201,158,140,96,77,120,211,148,130,86),(68,101,202,149,121,87,78,111,212,159,131,97),(69,112,203,160,122,98,79,102,213,150,132,88),(70,103,204,151,123,89,80,113,214,141,133,99)], [(1,123),(2,104),(3,125),(4,106),(5,127),(6,108),(7,129),(8,110),(9,131),(10,112),(11,133),(12,114),(13,135),(14,116),(15,137),(16,118),(17,139),(18,120),(19,121),(20,102),(21,67),(22,149),(23,69),(24,151),(25,71),(26,153),(27,73),(28,155),(29,75),(30,157),(31,77),(32,159),(33,79),(34,141),(35,61),(36,143),(37,63),(38,145),(39,65),(40,147),(41,89),(42,205),(43,91),(44,207),(45,93),(46,209),(47,95),(48,211),(49,97),(50,213),(51,99),(52,215),(53,81),(54,217),(55,83),(56,219),(57,85),(58,201),(59,87),(60,203),(62,233),(64,235),(66,237),(68,239),(70,221),(72,223),(74,225),(76,227),(78,229),(80,231),(82,174),(84,176),(86,178),(88,180),(90,162),(92,164),(94,166),(96,168),(98,170),(100,172),(101,191),(103,193),(105,195),(107,197),(109,199),(111,181),(113,183),(115,185),(117,187),(119,189),(122,182),(124,184),(126,186),(128,188),(130,190),(132,192),(134,194),(136,196),(138,198),(140,200),(142,222),(144,224),(146,226),(148,228),(150,230),(152,232),(154,234),(156,236),(158,238),(160,240),(161,214),(163,216),(165,218),(167,220),(169,202),(171,204),(173,206),(175,208),(177,210),(179,212)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,50,51,60),(42,59,52,49),(43,48,53,58),(44,57,54,47),(45,46,55,56),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70),(81,96,91,86),(82,85,92,95),(83,94,93,84),(87,90,97,100),(88,99,98,89),(101,104,111,114),(102,113,112,103),(105,120,115,110),(106,109,116,119),(107,118,117,108),(121,124,131,134),(122,133,132,123),(125,140,135,130),(126,129,136,139),(127,138,137,128),(141,160,151,150),(142,149,152,159),(143,158,153,148),(144,147,154,157),(145,156,155,146),(161,170,171,180),(162,179,172,169),(163,168,173,178),(164,177,174,167),(165,166,175,176),(181,194,191,184),(182,183,192,193),(185,190,195,200),(186,199,196,189),(187,188,197,198),(201,216,211,206),(202,205,212,215),(203,214,213,204),(207,210,217,220),(208,219,218,209),(221,230,231,240),(222,239,232,229),(223,228,233,238),(224,237,234,227),(225,226,235,236)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 12 | 2 | 2 | 4 | 10 | 20 | 60 | 2 | 2 | 2 | 10 | 10 | 12 | 60 | 2 | 2 | 24 | 24 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
45 irreducible representations
| dim | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C8.C22 | S3×D5 | D4×D5 | Q8.11D6 | C2×S3×D5 | SD16⋊D5 | D5×C3⋊D4 | D12.27D10 |
| kernel | D12.27D10 | C20.32D6 | C20.D6 | C3⋊Dic20 | C5×Q8⋊2S3 | C15⋊7Q16 | D12⋊5D5 | C3×Q8×D5 | Q8×D5 | C3×Dic5 | C6×D5 | Q8⋊2S3 | Dic10 | C4×D5 | C5×Q8 | C3⋊C8 | D12 | C3×Q8 | Dic5 | D10 | C15 | Q8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12.27D10 ►in GL6(𝔽241)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 69 | 76 |
| 0 | 0 | 0 | 240 | 69 | 69 |
| 0 | 0 | 138 | 89 | 1 | 0 |
| 0 | 0 | 103 | 138 | 0 | 1 |
| 102 | 225 | 0 | 0 | 0 | 0 |
| 123 | 139 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 226 | 88 | 238 |
| 0 | 0 | 15 | 220 | 153 | 88 |
| 0 | 0 | 40 | 89 | 21 | 15 |
| 0 | 0 | 40 | 40 | 226 | 220 |
| 171 | 101 | 0 | 0 | 0 | 0 |
| 140 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 204 | 204 | 44 | 158 |
| 0 | 0 | 37 | 4 | 158 | 44 |
| 0 | 0 | 51 | 52 | 237 | 37 |
| 0 | 0 | 239 | 51 | 204 | 37 |
| 70 | 140 | 0 | 0 | 0 | 0 |
| 101 | 171 | 0 | 0 | 0 | 0 |
| 0 | 0 | 204 | 204 | 44 | 158 |
| 0 | 0 | 4 | 37 | 158 | 61 |
| 0 | 0 | 52 | 51 | 237 | 0 |
| 0 | 0 | 51 | 239 | 204 | 4 |
G:=sub<GL(6,GF(241))| [1,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,138,103,0,0,0,240,89,138,0,0,69,69,1,0,0,0,76,69,0,1],[102,123,0,0,0,0,225,139,0,0,0,0,0,0,21,15,40,40,0,0,226,220,89,40,0,0,88,153,21,226,0,0,238,88,15,220],[171,140,0,0,0,0,101,70,0,0,0,0,0,0,204,37,51,239,0,0,204,4,52,51,0,0,44,158,237,204,0,0,158,44,37,37],[70,101,0,0,0,0,140,171,0,0,0,0,0,0,204,4,52,51,0,0,204,37,51,239,0,0,44,158,237,204,0,0,158,61,0,4] >;
D12.27D10 in GAP, Magma, Sage, TeX
D_{12}._{27}D_{10} % in TeX
G:=Group("D12.27D10"); // GroupNames label
G:=SmallGroup(480,589);
// by ID
G=gap.SmallGroup(480,589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,100,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^10=d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=c^9>;
// generators/relations