metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.43D4, C20.52D12, D12.33D10, C60.134C23, Dic6.35D10, Dic30.54C22, C5⋊2C8.4D6, C4○D12.5D5, (C2×C10).7D12, (C2×C30).65D4, C30.97(C2×D4), C5⋊5(C8.D6), C5⋊Dic12⋊14C2, C10.55(C2×D12), (C2×C20).104D6, C4.Dic5⋊10S3, (C2×Dic30)⋊19C2, (C2×C12).105D10, D12.D5⋊14C2, C3⋊1(D4.9D10), C4.17(C5⋊D12), C12.34(C5⋊D4), C15⋊12(C8.C22), (C2×C60).96C22, C20.101(C22×S3), (C5×D12).38C22, C12.157(C22×D5), C22.10(C5⋊D12), (C5×Dic6).40C22, C4.82(C2×S3×D5), C6.9(C2×C5⋊D4), (C2×C4).18(S3×D5), (C5×C4○D12).4C2, C2.13(C2×C5⋊D12), (C3×C4.Dic5)⋊9C2, (C2×C6).15(C5⋊D4), (C3×C5⋊2C8).22C22, SmallGroup(480,398)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.33D10
G = < a,b,c,d | a12=b2=1, c10=d2=a6, bab=cac-1=dad-1=a-1, cbc-1=a10b, dbd-1=ab, dcd-1=a9c9 >
Subgroups: 572 in 120 conjugacy classes, 44 normal (34 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, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C8.C22, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3, Dic15, C60, S3×C10, C2×C30, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, C8.D6, C3×C5⋊2C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, Dic30, C2×Dic15, C2×C60, D4.9D10, D12.D5, C5⋊Dic12, C3×C4.Dic5, C5×C4○D12, C2×Dic30, D12.33D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8.C22, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8.D6, C5⋊D12, C2×S3×D5, D4.9D10, C2×C5⋊D12, D12.33D10
►On 240 points
Generators in S240
(1 96 123 72 197 177 11 86 133 62 187 167)(2 168 188 63 134 87 12 178 198 73 124 97)(3 98 125 74 199 179 13 88 135 64 189 169)(4 170 190 65 136 89 14 180 200 75 126 99)(5 100 127 76 181 161 15 90 137 66 191 171)(6 172 192 67 138 91 16 162 182 77 128 81)(7 82 129 78 183 163 17 92 139 68 193 173)(8 174 194 69 140 93 18 164 184 79 130 83)(9 84 131 80 185 165 19 94 121 70 195 175)(10 176 196 71 122 95 20 166 186 61 132 85)(21 227 119 218 60 152 31 237 109 208 50 142)(22 143 51 209 110 238 32 153 41 219 120 228)(23 229 101 220 42 154 33 239 111 210 52 144)(24 145 53 211 112 240 34 155 43 201 102 230)(25 231 103 202 44 156 35 221 113 212 54 146)(26 147 55 213 114 222 36 157 45 203 104 232)(27 233 105 204 46 158 37 223 115 214 56 148)(28 149 57 215 116 224 38 159 47 205 106 234)(29 235 107 206 48 160 39 225 117 216 58 150)(30 151 59 217 118 226 40 141 49 207 108 236)
(1 182)(2 129)(3 184)(4 131)(5 186)(6 133)(7 188)(8 135)(9 190)(10 137)(11 192)(12 139)(13 194)(14 121)(15 196)(16 123)(17 198)(18 125)(19 200)(20 127)(21 45)(22 105)(23 47)(24 107)(25 49)(26 109)(27 51)(28 111)(29 53)(30 113)(31 55)(32 115)(33 57)(34 117)(35 59)(36 119)(37 41)(38 101)(39 43)(40 103)(42 116)(44 118)(46 120)(48 102)(50 104)(52 106)(54 108)(56 110)(58 112)(60 114)(61 171)(62 81)(63 173)(64 83)(65 175)(66 85)(67 177)(68 87)(69 179)(70 89)(71 161)(72 91)(73 163)(74 93)(75 165)(76 95)(77 167)(78 97)(79 169)(80 99)(82 168)(84 170)(86 172)(88 174)(90 176)(92 178)(94 180)(96 162)(98 164)(100 166)(122 181)(124 183)(126 185)(128 187)(130 189)(132 191)(134 193)(136 195)(138 197)(140 199)(141 231)(142 203)(143 233)(144 205)(145 235)(146 207)(147 237)(148 209)(149 239)(150 211)(151 221)(152 213)(153 223)(154 215)(155 225)(156 217)(157 227)(158 219)(159 229)(160 201)(202 226)(204 228)(206 230)(208 232)(210 234)(212 236)(214 238)(216 240)(218 222)(220 224)
(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)
(1 152 11 142)(2 108 12 118)(3 150 13 160)(4 106 14 116)(5 148 15 158)(6 104 16 114)(7 146 17 156)(8 102 18 112)(9 144 19 154)(10 120 20 110)(21 167 31 177)(22 122 32 132)(23 165 33 175)(24 140 34 130)(25 163 35 173)(26 138 36 128)(27 161 37 171)(28 136 38 126)(29 179 39 169)(30 134 40 124)(41 186 51 196)(42 84 52 94)(43 184 53 194)(44 82 54 92)(45 182 55 192)(46 100 56 90)(47 200 57 190)(48 98 58 88)(49 198 59 188)(50 96 60 86)(61 143 71 153)(62 109 72 119)(63 141 73 151)(64 107 74 117)(65 159 75 149)(66 105 76 115)(67 157 77 147)(68 103 78 113)(69 155 79 145)(70 101 80 111)(81 232 91 222)(83 230 93 240)(85 228 95 238)(87 226 97 236)(89 224 99 234)(121 220 131 210)(123 218 133 208)(125 216 135 206)(127 214 137 204)(129 212 139 202)(162 213 172 203)(164 211 174 201)(166 209 176 219)(168 207 178 217)(170 205 180 215)(181 223 191 233)(183 221 193 231)(185 239 195 229)(187 237 197 227)(189 235 199 225)
G:=sub<Sym(240)| (1,96,123,72,197,177,11,86,133,62,187,167)(2,168,188,63,134,87,12,178,198,73,124,97)(3,98,125,74,199,179,13,88,135,64,189,169)(4,170,190,65,136,89,14,180,200,75,126,99)(5,100,127,76,181,161,15,90,137,66,191,171)(6,172,192,67,138,91,16,162,182,77,128,81)(7,82,129,78,183,163,17,92,139,68,193,173)(8,174,194,69,140,93,18,164,184,79,130,83)(9,84,131,80,185,165,19,94,121,70,195,175)(10,176,196,71,122,95,20,166,186,61,132,85)(21,227,119,218,60,152,31,237,109,208,50,142)(22,143,51,209,110,238,32,153,41,219,120,228)(23,229,101,220,42,154,33,239,111,210,52,144)(24,145,53,211,112,240,34,155,43,201,102,230)(25,231,103,202,44,156,35,221,113,212,54,146)(26,147,55,213,114,222,36,157,45,203,104,232)(27,233,105,204,46,158,37,223,115,214,56,148)(28,149,57,215,116,224,38,159,47,205,106,234)(29,235,107,206,48,160,39,225,117,216,58,150)(30,151,59,217,118,226,40,141,49,207,108,236), (1,182)(2,129)(3,184)(4,131)(5,186)(6,133)(7,188)(8,135)(9,190)(10,137)(11,192)(12,139)(13,194)(14,121)(15,196)(16,123)(17,198)(18,125)(19,200)(20,127)(21,45)(22,105)(23,47)(24,107)(25,49)(26,109)(27,51)(28,111)(29,53)(30,113)(31,55)(32,115)(33,57)(34,117)(35,59)(36,119)(37,41)(38,101)(39,43)(40,103)(42,116)(44,118)(46,120)(48,102)(50,104)(52,106)(54,108)(56,110)(58,112)(60,114)(61,171)(62,81)(63,173)(64,83)(65,175)(66,85)(67,177)(68,87)(69,179)(70,89)(71,161)(72,91)(73,163)(74,93)(75,165)(76,95)(77,167)(78,97)(79,169)(80,99)(82,168)(84,170)(86,172)(88,174)(90,176)(92,178)(94,180)(96,162)(98,164)(100,166)(122,181)(124,183)(126,185)(128,187)(130,189)(132,191)(134,193)(136,195)(138,197)(140,199)(141,231)(142,203)(143,233)(144,205)(145,235)(146,207)(147,237)(148,209)(149,239)(150,211)(151,221)(152,213)(153,223)(154,215)(155,225)(156,217)(157,227)(158,219)(159,229)(160,201)(202,226)(204,228)(206,230)(208,232)(210,234)(212,236)(214,238)(216,240)(218,222)(220,224), (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), (1,152,11,142)(2,108,12,118)(3,150,13,160)(4,106,14,116)(5,148,15,158)(6,104,16,114)(7,146,17,156)(8,102,18,112)(9,144,19,154)(10,120,20,110)(21,167,31,177)(22,122,32,132)(23,165,33,175)(24,140,34,130)(25,163,35,173)(26,138,36,128)(27,161,37,171)(28,136,38,126)(29,179,39,169)(30,134,40,124)(41,186,51,196)(42,84,52,94)(43,184,53,194)(44,82,54,92)(45,182,55,192)(46,100,56,90)(47,200,57,190)(48,98,58,88)(49,198,59,188)(50,96,60,86)(61,143,71,153)(62,109,72,119)(63,141,73,151)(64,107,74,117)(65,159,75,149)(66,105,76,115)(67,157,77,147)(68,103,78,113)(69,155,79,145)(70,101,80,111)(81,232,91,222)(83,230,93,240)(85,228,95,238)(87,226,97,236)(89,224,99,234)(121,220,131,210)(123,218,133,208)(125,216,135,206)(127,214,137,204)(129,212,139,202)(162,213,172,203)(164,211,174,201)(166,209,176,219)(168,207,178,217)(170,205,180,215)(181,223,191,233)(183,221,193,231)(185,239,195,229)(187,237,197,227)(189,235,199,225)>;
G:=Group( (1,96,123,72,197,177,11,86,133,62,187,167)(2,168,188,63,134,87,12,178,198,73,124,97)(3,98,125,74,199,179,13,88,135,64,189,169)(4,170,190,65,136,89,14,180,200,75,126,99)(5,100,127,76,181,161,15,90,137,66,191,171)(6,172,192,67,138,91,16,162,182,77,128,81)(7,82,129,78,183,163,17,92,139,68,193,173)(8,174,194,69,140,93,18,164,184,79,130,83)(9,84,131,80,185,165,19,94,121,70,195,175)(10,176,196,71,122,95,20,166,186,61,132,85)(21,227,119,218,60,152,31,237,109,208,50,142)(22,143,51,209,110,238,32,153,41,219,120,228)(23,229,101,220,42,154,33,239,111,210,52,144)(24,145,53,211,112,240,34,155,43,201,102,230)(25,231,103,202,44,156,35,221,113,212,54,146)(26,147,55,213,114,222,36,157,45,203,104,232)(27,233,105,204,46,158,37,223,115,214,56,148)(28,149,57,215,116,224,38,159,47,205,106,234)(29,235,107,206,48,160,39,225,117,216,58,150)(30,151,59,217,118,226,40,141,49,207,108,236), (1,182)(2,129)(3,184)(4,131)(5,186)(6,133)(7,188)(8,135)(9,190)(10,137)(11,192)(12,139)(13,194)(14,121)(15,196)(16,123)(17,198)(18,125)(19,200)(20,127)(21,45)(22,105)(23,47)(24,107)(25,49)(26,109)(27,51)(28,111)(29,53)(30,113)(31,55)(32,115)(33,57)(34,117)(35,59)(36,119)(37,41)(38,101)(39,43)(40,103)(42,116)(44,118)(46,120)(48,102)(50,104)(52,106)(54,108)(56,110)(58,112)(60,114)(61,171)(62,81)(63,173)(64,83)(65,175)(66,85)(67,177)(68,87)(69,179)(70,89)(71,161)(72,91)(73,163)(74,93)(75,165)(76,95)(77,167)(78,97)(79,169)(80,99)(82,168)(84,170)(86,172)(88,174)(90,176)(92,178)(94,180)(96,162)(98,164)(100,166)(122,181)(124,183)(126,185)(128,187)(130,189)(132,191)(134,193)(136,195)(138,197)(140,199)(141,231)(142,203)(143,233)(144,205)(145,235)(146,207)(147,237)(148,209)(149,239)(150,211)(151,221)(152,213)(153,223)(154,215)(155,225)(156,217)(157,227)(158,219)(159,229)(160,201)(202,226)(204,228)(206,230)(208,232)(210,234)(212,236)(214,238)(216,240)(218,222)(220,224), (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), (1,152,11,142)(2,108,12,118)(3,150,13,160)(4,106,14,116)(5,148,15,158)(6,104,16,114)(7,146,17,156)(8,102,18,112)(9,144,19,154)(10,120,20,110)(21,167,31,177)(22,122,32,132)(23,165,33,175)(24,140,34,130)(25,163,35,173)(26,138,36,128)(27,161,37,171)(28,136,38,126)(29,179,39,169)(30,134,40,124)(41,186,51,196)(42,84,52,94)(43,184,53,194)(44,82,54,92)(45,182,55,192)(46,100,56,90)(47,200,57,190)(48,98,58,88)(49,198,59,188)(50,96,60,86)(61,143,71,153)(62,109,72,119)(63,141,73,151)(64,107,74,117)(65,159,75,149)(66,105,76,115)(67,157,77,147)(68,103,78,113)(69,155,79,145)(70,101,80,111)(81,232,91,222)(83,230,93,240)(85,228,95,238)(87,226,97,236)(89,224,99,234)(121,220,131,210)(123,218,133,208)(125,216,135,206)(127,214,137,204)(129,212,139,202)(162,213,172,203)(164,211,174,201)(166,209,176,219)(168,207,178,217)(170,205,180,215)(181,223,191,233)(183,221,193,231)(185,239,195,229)(187,237,197,227)(189,235,199,225) );
G=PermutationGroup([[(1,96,123,72,197,177,11,86,133,62,187,167),(2,168,188,63,134,87,12,178,198,73,124,97),(3,98,125,74,199,179,13,88,135,64,189,169),(4,170,190,65,136,89,14,180,200,75,126,99),(5,100,127,76,181,161,15,90,137,66,191,171),(6,172,192,67,138,91,16,162,182,77,128,81),(7,82,129,78,183,163,17,92,139,68,193,173),(8,174,194,69,140,93,18,164,184,79,130,83),(9,84,131,80,185,165,19,94,121,70,195,175),(10,176,196,71,122,95,20,166,186,61,132,85),(21,227,119,218,60,152,31,237,109,208,50,142),(22,143,51,209,110,238,32,153,41,219,120,228),(23,229,101,220,42,154,33,239,111,210,52,144),(24,145,53,211,112,240,34,155,43,201,102,230),(25,231,103,202,44,156,35,221,113,212,54,146),(26,147,55,213,114,222,36,157,45,203,104,232),(27,233,105,204,46,158,37,223,115,214,56,148),(28,149,57,215,116,224,38,159,47,205,106,234),(29,235,107,206,48,160,39,225,117,216,58,150),(30,151,59,217,118,226,40,141,49,207,108,236)], [(1,182),(2,129),(3,184),(4,131),(5,186),(6,133),(7,188),(8,135),(9,190),(10,137),(11,192),(12,139),(13,194),(14,121),(15,196),(16,123),(17,198),(18,125),(19,200),(20,127),(21,45),(22,105),(23,47),(24,107),(25,49),(26,109),(27,51),(28,111),(29,53),(30,113),(31,55),(32,115),(33,57),(34,117),(35,59),(36,119),(37,41),(38,101),(39,43),(40,103),(42,116),(44,118),(46,120),(48,102),(50,104),(52,106),(54,108),(56,110),(58,112),(60,114),(61,171),(62,81),(63,173),(64,83),(65,175),(66,85),(67,177),(68,87),(69,179),(70,89),(71,161),(72,91),(73,163),(74,93),(75,165),(76,95),(77,167),(78,97),(79,169),(80,99),(82,168),(84,170),(86,172),(88,174),(90,176),(92,178),(94,180),(96,162),(98,164),(100,166),(122,181),(124,183),(126,185),(128,187),(130,189),(132,191),(134,193),(136,195),(138,197),(140,199),(141,231),(142,203),(143,233),(144,205),(145,235),(146,207),(147,237),(148,209),(149,239),(150,211),(151,221),(152,213),(153,223),(154,215),(155,225),(156,217),(157,227),(158,219),(159,229),(160,201),(202,226),(204,228),(206,230),(208,232),(210,234),(212,236),(214,238),(216,240),(218,222),(220,224)], [(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)], [(1,152,11,142),(2,108,12,118),(3,150,13,160),(4,106,14,116),(5,148,15,158),(6,104,16,114),(7,146,17,156),(8,102,18,112),(9,144,19,154),(10,120,20,110),(21,167,31,177),(22,122,32,132),(23,165,33,175),(24,140,34,130),(25,163,35,173),(26,138,36,128),(27,161,37,171),(28,136,38,126),(29,179,39,169),(30,134,40,124),(41,186,51,196),(42,84,52,94),(43,184,53,194),(44,82,54,92),(45,182,55,192),(46,100,56,90),(47,200,57,190),(48,98,58,88),(49,198,59,188),(50,96,60,86),(61,143,71,153),(62,109,72,119),(63,141,73,151),(64,107,74,117),(65,159,75,149),(66,105,76,115),(67,157,77,147),(68,103,78,113),(69,155,79,145),(70,101,80,111),(81,232,91,222),(83,230,93,240),(85,228,95,238),(87,226,97,236),(89,224,99,234),(121,220,131,210),(123,218,133,208),(125,216,135,206),(127,214,137,204),(129,212,139,202),(162,213,172,203),(164,211,174,201),(166,209,176,219),(168,207,178,217),(170,205,180,215),(181,223,191,233),(183,221,193,231),(185,239,195,229),(187,237,197,227),(189,235,199,225)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 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 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 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 | 12 | 2 | 2 | 2 | 12 | 60 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 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 |
57 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D5 | C8.D6 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | D4.9D10 | D12.33D10 |
| kernel | D12.33D10 | D12.D5 | C5⋊Dic12 | C3×C4.Dic5 | C5×C4○D12 | C2×Dic30 | C4.Dic5 | C60 | C2×C30 | C4○D12 | C5⋊2C8 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D12.33D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 142 | 0 | 0 |
| 0 | 0 | 99 | 198 | 0 | 0 |
| 0 | 0 | 0 | 0 | 198 | 99 |
| 0 | 0 | 0 | 0 | 142 | 99 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 | 0 |
| 189 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 35 | 54 | 0 | 0 | 0 | 0 |
| 138 | 206 | 0 | 0 | 0 | 0 |
| 0 | 0 | 211 | 8 | 0 | 0 |
| 0 | 0 | 38 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 216 | 225 |
| 0 | 0 | 0 | 0 | 9 | 25 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,99,99,0,0,0,0,142,198,0,0,0,0,0,0,198,142,0,0,0,0,99,99],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,0,1,0,0,0,0,240,1,0,0],[0,189,0,0,0,0,51,52,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[35,138,0,0,0,0,54,206,0,0,0,0,0,0,211,38,0,0,0,0,8,30,0,0,0,0,0,0,216,9,0,0,0,0,225,25] >;
D12.33D10 in GAP, Magma, Sage, TeX
D_{12}._{33}D_{10} % in TeX
G:=Group("D12.33D10"); // GroupNames label
G:=SmallGroup(480,398);
// by ID
G=gap.SmallGroup(480,398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,100,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^10=d^2=a^6,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^10*b,d*b*d^-1=a*b,d*c*d^-1=a^9*c^9>;
// generators/relations