metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.16D6, D30.14D4, C60.45C23, Dic6.16D10, Dic15.46D4, D60.17C22, Q8⋊D5⋊5S3, C3⋊D40⋊8C2, C3⋊C8.21D10, C3⋊Q16⋊5D5, C6.79(D4×D5), C15⋊23(C4○D8), C10.80(S3×D4), C5⋊2C8.21D6, Q8.22(S3×D5), (C3×Q8).9D10, (C5×Q8).26D6, Q8⋊3D15⋊2C2, D20⋊S3⋊3C2, D15⋊2C8⋊6C2, C3⋊3(Q8.D10), Dic6⋊D5⋊8C2, C30.207(C2×D4), C5⋊4(Q8.7D6), C20.45(C22×S3), C12.45(C22×D5), (C4×D15).13C22, (C3×D20).17C22, (Q8×C15).15C22, C2.32(D10⋊D6), (C5×Dic6).17C22, C4.45(C2×S3×D5), (C3×Q8⋊D5)⋊7C2, (C5×C3⋊Q16)⋊7C2, (C5×C3⋊C8).15C22, (C3×C5⋊2C8).15C22, SmallGroup(480,597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.16D6
G = < a,b,c,d | a30=b2=d2=1, c4=a15, bab=a-1, cac-1=dad=a19, cbc-1=a18b, dbd=a3b, dcd=c3 >
Subgroups: 796 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15, C30, C4○D8, C5⋊2C8, C40, C4×D5, D20, D20, C5×Q8, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, Dic15, C60, C60, C6×D5, D30, D30, C8×D5, D40, Q8⋊D5, Q8⋊D5, C5×Q16, Q8⋊2D5, Q8.7D6, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, C4×D15, D60, D60, Q8×C15, Q8.D10, D15⋊2C8, C3⋊D40, Dic6⋊D5, C3×Q8⋊D5, C5×C3⋊Q16, D20⋊S3, Q8⋊3D15, D20.16D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, Q8.7D6, C2×S3×D5, Q8.D10, D10⋊D6, D20.16D6
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 90)(87 89)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(146 150)(147 149)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 220)(212 219)(213 218)(214 217)(215 216)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)
(1 216 51 186 16 231 36 201)(2 235 52 205 17 220 37 190)(3 224 53 194 18 239 38 209)(4 213 54 183 19 228 39 198)(5 232 55 202 20 217 40 187)(6 221 56 191 21 236 41 206)(7 240 57 210 22 225 42 195)(8 229 58 199 23 214 43 184)(9 218 59 188 24 233 44 203)(10 237 60 207 25 222 45 192)(11 226 31 196 26 211 46 181)(12 215 32 185 27 230 47 200)(13 234 33 204 28 219 48 189)(14 223 34 193 29 238 49 208)(15 212 35 182 30 227 50 197)(61 161 101 136 76 176 116 121)(62 180 102 125 77 165 117 140)(63 169 103 144 78 154 118 129)(64 158 104 133 79 173 119 148)(65 177 105 122 80 162 120 137)(66 166 106 141 81 151 91 126)(67 155 107 130 82 170 92 145)(68 174 108 149 83 159 93 134)(69 163 109 138 84 178 94 123)(70 152 110 127 85 167 95 142)(71 171 111 146 86 156 96 131)(72 160 112 135 87 175 97 150)(73 179 113 124 88 164 98 139)(74 168 114 143 89 153 99 128)(75 157 115 132 90 172 100 147)
(1 126)(2 145)(3 134)(4 123)(5 142)(6 131)(7 150)(8 139)(9 128)(10 147)(11 136)(12 125)(13 144)(14 133)(15 122)(16 141)(17 130)(18 149)(19 138)(20 127)(21 146)(22 135)(23 124)(24 143)(25 132)(26 121)(27 140)(28 129)(29 148)(30 137)(31 161)(32 180)(33 169)(34 158)(35 177)(36 166)(37 155)(38 174)(39 163)(40 152)(41 171)(42 160)(43 179)(44 168)(45 157)(46 176)(47 165)(48 154)(49 173)(50 162)(51 151)(52 170)(53 159)(54 178)(55 167)(56 156)(57 175)(58 164)(59 153)(60 172)(61 181)(62 200)(63 189)(64 208)(65 197)(66 186)(67 205)(68 194)(69 183)(70 202)(71 191)(72 210)(73 199)(74 188)(75 207)(76 196)(77 185)(78 204)(79 193)(80 182)(81 201)(82 190)(83 209)(84 198)(85 187)(86 206)(87 195)(88 184)(89 203)(90 192)(91 231)(92 220)(93 239)(94 228)(95 217)(96 236)(97 225)(98 214)(99 233)(100 222)(101 211)(102 230)(103 219)(104 238)(105 227)(106 216)(107 235)(108 224)(109 213)(110 232)(111 221)(112 240)(113 229)(114 218)(115 237)(116 226)(117 215)(118 234)(119 223)(120 212)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,220)(212,219)(213,218)(214,217)(215,216)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,216,51,186,16,231,36,201)(2,235,52,205,17,220,37,190)(3,224,53,194,18,239,38,209)(4,213,54,183,19,228,39,198)(5,232,55,202,20,217,40,187)(6,221,56,191,21,236,41,206)(7,240,57,210,22,225,42,195)(8,229,58,199,23,214,43,184)(9,218,59,188,24,233,44,203)(10,237,60,207,25,222,45,192)(11,226,31,196,26,211,46,181)(12,215,32,185,27,230,47,200)(13,234,33,204,28,219,48,189)(14,223,34,193,29,238,49,208)(15,212,35,182,30,227,50,197)(61,161,101,136,76,176,116,121)(62,180,102,125,77,165,117,140)(63,169,103,144,78,154,118,129)(64,158,104,133,79,173,119,148)(65,177,105,122,80,162,120,137)(66,166,106,141,81,151,91,126)(67,155,107,130,82,170,92,145)(68,174,108,149,83,159,93,134)(69,163,109,138,84,178,94,123)(70,152,110,127,85,167,95,142)(71,171,111,146,86,156,96,131)(72,160,112,135,87,175,97,150)(73,179,113,124,88,164,98,139)(74,168,114,143,89,153,99,128)(75,157,115,132,90,172,100,147), (1,126)(2,145)(3,134)(4,123)(5,142)(6,131)(7,150)(8,139)(9,128)(10,147)(11,136)(12,125)(13,144)(14,133)(15,122)(16,141)(17,130)(18,149)(19,138)(20,127)(21,146)(22,135)(23,124)(24,143)(25,132)(26,121)(27,140)(28,129)(29,148)(30,137)(31,161)(32,180)(33,169)(34,158)(35,177)(36,166)(37,155)(38,174)(39,163)(40,152)(41,171)(42,160)(43,179)(44,168)(45,157)(46,176)(47,165)(48,154)(49,173)(50,162)(51,151)(52,170)(53,159)(54,178)(55,167)(56,156)(57,175)(58,164)(59,153)(60,172)(61,181)(62,200)(63,189)(64,208)(65,197)(66,186)(67,205)(68,194)(69,183)(70,202)(71,191)(72,210)(73,199)(74,188)(75,207)(76,196)(77,185)(78,204)(79,193)(80,182)(81,201)(82,190)(83,209)(84,198)(85,187)(86,206)(87,195)(88,184)(89,203)(90,192)(91,231)(92,220)(93,239)(94,228)(95,217)(96,236)(97,225)(98,214)(99,233)(100,222)(101,211)(102,230)(103,219)(104,238)(105,227)(106,216)(107,235)(108,224)(109,213)(110,232)(111,221)(112,240)(113,229)(114,218)(115,237)(116,226)(117,215)(118,234)(119,223)(120,212)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,220)(212,219)(213,218)(214,217)(215,216)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,216,51,186,16,231,36,201)(2,235,52,205,17,220,37,190)(3,224,53,194,18,239,38,209)(4,213,54,183,19,228,39,198)(5,232,55,202,20,217,40,187)(6,221,56,191,21,236,41,206)(7,240,57,210,22,225,42,195)(8,229,58,199,23,214,43,184)(9,218,59,188,24,233,44,203)(10,237,60,207,25,222,45,192)(11,226,31,196,26,211,46,181)(12,215,32,185,27,230,47,200)(13,234,33,204,28,219,48,189)(14,223,34,193,29,238,49,208)(15,212,35,182,30,227,50,197)(61,161,101,136,76,176,116,121)(62,180,102,125,77,165,117,140)(63,169,103,144,78,154,118,129)(64,158,104,133,79,173,119,148)(65,177,105,122,80,162,120,137)(66,166,106,141,81,151,91,126)(67,155,107,130,82,170,92,145)(68,174,108,149,83,159,93,134)(69,163,109,138,84,178,94,123)(70,152,110,127,85,167,95,142)(71,171,111,146,86,156,96,131)(72,160,112,135,87,175,97,150)(73,179,113,124,88,164,98,139)(74,168,114,143,89,153,99,128)(75,157,115,132,90,172,100,147), (1,126)(2,145)(3,134)(4,123)(5,142)(6,131)(7,150)(8,139)(9,128)(10,147)(11,136)(12,125)(13,144)(14,133)(15,122)(16,141)(17,130)(18,149)(19,138)(20,127)(21,146)(22,135)(23,124)(24,143)(25,132)(26,121)(27,140)(28,129)(29,148)(30,137)(31,161)(32,180)(33,169)(34,158)(35,177)(36,166)(37,155)(38,174)(39,163)(40,152)(41,171)(42,160)(43,179)(44,168)(45,157)(46,176)(47,165)(48,154)(49,173)(50,162)(51,151)(52,170)(53,159)(54,178)(55,167)(56,156)(57,175)(58,164)(59,153)(60,172)(61,181)(62,200)(63,189)(64,208)(65,197)(66,186)(67,205)(68,194)(69,183)(70,202)(71,191)(72,210)(73,199)(74,188)(75,207)(76,196)(77,185)(78,204)(79,193)(80,182)(81,201)(82,190)(83,209)(84,198)(85,187)(86,206)(87,195)(88,184)(89,203)(90,192)(91,231)(92,220)(93,239)(94,228)(95,217)(96,236)(97,225)(98,214)(99,233)(100,222)(101,211)(102,230)(103,219)(104,238)(105,227)(106,216)(107,235)(108,224)(109,213)(110,232)(111,221)(112,240)(113,229)(114,218)(115,237)(116,226)(117,215)(118,234)(119,223)(120,212) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,90),(87,89),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(146,150),(147,149),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(181,190),(182,189),(183,188),(184,187),(185,186),(191,210),(192,209),(193,208),(194,207),(195,206),(196,205),(197,204),(198,203),(199,202),(200,201),(211,220),(212,219),(213,218),(214,217),(215,216),(221,240),(222,239),(223,238),(224,237),(225,236),(226,235),(227,234),(228,233),(229,232),(230,231)], [(1,216,51,186,16,231,36,201),(2,235,52,205,17,220,37,190),(3,224,53,194,18,239,38,209),(4,213,54,183,19,228,39,198),(5,232,55,202,20,217,40,187),(6,221,56,191,21,236,41,206),(7,240,57,210,22,225,42,195),(8,229,58,199,23,214,43,184),(9,218,59,188,24,233,44,203),(10,237,60,207,25,222,45,192),(11,226,31,196,26,211,46,181),(12,215,32,185,27,230,47,200),(13,234,33,204,28,219,48,189),(14,223,34,193,29,238,49,208),(15,212,35,182,30,227,50,197),(61,161,101,136,76,176,116,121),(62,180,102,125,77,165,117,140),(63,169,103,144,78,154,118,129),(64,158,104,133,79,173,119,148),(65,177,105,122,80,162,120,137),(66,166,106,141,81,151,91,126),(67,155,107,130,82,170,92,145),(68,174,108,149,83,159,93,134),(69,163,109,138,84,178,94,123),(70,152,110,127,85,167,95,142),(71,171,111,146,86,156,96,131),(72,160,112,135,87,175,97,150),(73,179,113,124,88,164,98,139),(74,168,114,143,89,153,99,128),(75,157,115,132,90,172,100,147)], [(1,126),(2,145),(3,134),(4,123),(5,142),(6,131),(7,150),(8,139),(9,128),(10,147),(11,136),(12,125),(13,144),(14,133),(15,122),(16,141),(17,130),(18,149),(19,138),(20,127),(21,146),(22,135),(23,124),(24,143),(25,132),(26,121),(27,140),(28,129),(29,148),(30,137),(31,161),(32,180),(33,169),(34,158),(35,177),(36,166),(37,155),(38,174),(39,163),(40,152),(41,171),(42,160),(43,179),(44,168),(45,157),(46,176),(47,165),(48,154),(49,173),(50,162),(51,151),(52,170),(53,159),(54,178),(55,167),(56,156),(57,175),(58,164),(59,153),(60,172),(61,181),(62,200),(63,189),(64,208),(65,197),(66,186),(67,205),(68,194),(69,183),(70,202),(71,191),(72,210),(73,199),(74,188),(75,207),(76,196),(77,185),(78,204),(79,193),(80,182),(81,201),(82,190),(83,209),(84,198),(85,187),(86,206),(87,195),(88,184),(89,203),(90,192),(91,231),(92,220),(93,239),(94,228),(95,217),(96,236),(97,225),(98,214),(99,233),(100,222),(101,211),(102,230),(103,219),(104,238),(105,227),(106,216),(107,235),(108,224),(109,213),(110,232),(111,221),(112,240),(113,229),(114,218),(115,237),(116,226),(117,215),(118,234),(119,223),(120,212)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 20 | 30 | 60 | 2 | 2 | 4 | 12 | 15 | 15 | 2 | 2 | 2 | 40 | 6 | 6 | 10 | 10 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 20 | 20 | 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 | 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 | C4○D8 | S3×D4 | S3×D5 | D4×D5 | Q8.7D6 | C2×S3×D5 | Q8.D10 | D10⋊D6 | D20.16D6 |
| kernel | D20.16D6 | D15⋊2C8 | C3⋊D40 | Dic6⋊D5 | C3×Q8⋊D5 | C5×C3⋊Q16 | D20⋊S3 | Q8⋊3D15 | Q8⋊D5 | Dic15 | D30 | C3⋊Q16 | C5⋊2C8 | D20 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | C15 | C10 | 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 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.16D6 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 239 | 192 |
| 0 | 0 | 0 | 0 | 64 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 52 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 64 | 1 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,239,64,0,0,0,0,192,1],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,240,64,0,0,0,0,0,1],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D20.16D6 in GAP, Magma, Sage, TeX
D_{20}._{16}D_6 % in TeX
G:=Group("D20.16D6"); // GroupNames label
G:=SmallGroup(480,597);
// by ID
G=gap.SmallGroup(480,597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,100,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d=a^19,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations