metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.9D20, D6⋊C4⋊6D5, C60⋊5C4⋊7C2, (S3×C10).7D4, C6.22(C2×D20), C2.23(S3×D20), (C2×C20).28D6, C30.63(C2×D4), C10.21(S3×D4), D10⋊C4⋊5S3, (C2×C12).28D10, C5⋊3(C23.9D6), (C2×C60).13C22, C6.Dic10⋊26C2, (C22×D5).18D6, C10.62(C4○D12), C30.126(C4○D4), C6.31(D4⋊2D5), D10⋊Dic3⋊18C2, (C2×C30).147C23, (C2×Dic5).120D6, (C2×Dic3).45D10, (C22×S3).47D10, C3⋊2(C22.D20), C10.73(D4⋊2S3), C2.18(D12⋊5D5), C15⋊15(C22.D4), (C6×Dic5).88C22, C2.18(C30.C23), (C10×Dic3).89C22, (C2×Dic15).114C22, (C5×D6⋊C4)⋊6C2, (C2×S3×Dic5)⋊11C2, (C2×C4).61(S3×D5), (C2×C15⋊D4).7C2, (C3×D10⋊C4)⋊5C2, (D5×C2×C6).32C22, C22.199(C2×S3×D5), (S3×C2×C10).34C22, (C2×C6).159(C22×D5), (C2×C10).159(C22×S3), SmallGroup(480,533)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.9D20
G = < a,b,c,d | a6=b2=c20=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >
Subgroups: 812 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, S3×C10, C2×C30, C4⋊Dic5, D10⋊C4, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C23.9D6, S3×Dic5, C15⋊D4, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22.D20, D10⋊Dic3, C6.Dic10, C3×D10⋊C4, C5×D6⋊C4, C60⋊5C4, C2×S3×Dic5, C2×C15⋊D4, D6.9D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, D20, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, C2×D20, D4⋊2D5, C23.9D6, C2×S3×D5, C22.D20, D12⋊5D5, S3×D20, C30.C23, D6.9D20
►On 240 points
Generators in S240
(1 164 133 28 85 223)(2 165 134 29 86 224)(3 166 135 30 87 225)(4 167 136 31 88 226)(5 168 137 32 89 227)(6 169 138 33 90 228)(7 170 139 34 91 229)(8 171 140 35 92 230)(9 172 121 36 93 231)(10 173 122 37 94 232)(11 174 123 38 95 233)(12 175 124 39 96 234)(13 176 125 40 97 235)(14 177 126 21 98 236)(15 178 127 22 99 237)(16 179 128 23 100 238)(17 180 129 24 81 239)(18 161 130 25 82 240)(19 162 131 26 83 221)(20 163 132 27 84 222)(41 157 215 108 183 62)(42 158 216 109 184 63)(43 159 217 110 185 64)(44 160 218 111 186 65)(45 141 219 112 187 66)(46 142 220 113 188 67)(47 143 201 114 189 68)(48 144 202 115 190 69)(49 145 203 116 191 70)(50 146 204 117 192 71)(51 147 205 118 193 72)(52 148 206 119 194 73)(53 149 207 120 195 74)(54 150 208 101 196 75)(55 151 209 102 197 76)(56 152 210 103 198 77)(57 153 211 104 199 78)(58 154 212 105 200 79)(59 155 213 106 181 80)(60 156 214 107 182 61)
(1 223)(2 134)(3 225)(4 136)(5 227)(6 138)(7 229)(8 140)(9 231)(10 122)(11 233)(12 124)(13 235)(14 126)(15 237)(16 128)(17 239)(18 130)(19 221)(20 132)(21 236)(22 127)(23 238)(24 129)(25 240)(26 131)(27 222)(28 133)(29 224)(30 135)(31 226)(32 137)(33 228)(34 139)(35 230)(36 121)(37 232)(38 123)(39 234)(40 125)(41 108)(43 110)(45 112)(47 114)(49 116)(51 118)(53 120)(55 102)(57 104)(59 106)(61 156)(62 183)(63 158)(64 185)(65 160)(66 187)(67 142)(68 189)(69 144)(70 191)(71 146)(72 193)(73 148)(74 195)(75 150)(76 197)(77 152)(78 199)(79 154)(80 181)(81 180)(83 162)(85 164)(87 166)(89 168)(91 170)(93 172)(95 174)(97 176)(99 178)(141 219)(143 201)(145 203)(147 205)(149 207)(151 209)(153 211)(155 213)(157 215)(159 217)(182 214)(184 216)(186 218)(188 220)(190 202)(192 204)(194 206)(196 208)(198 210)(200 212)
(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)
(1 207 28 74)(2 73 29 206)(3 205 30 72)(4 71 31 204)(5 203 32 70)(6 69 33 202)(7 201 34 68)(8 67 35 220)(9 219 36 66)(10 65 37 218)(11 217 38 64)(12 63 39 216)(13 215 40 62)(14 61 21 214)(15 213 22 80)(16 79 23 212)(17 211 24 78)(18 77 25 210)(19 209 26 76)(20 75 27 208)(41 176 108 97)(42 96 109 175)(43 174 110 95)(44 94 111 173)(45 172 112 93)(46 92 113 171)(47 170 114 91)(48 90 115 169)(49 168 116 89)(50 88 117 167)(51 166 118 87)(52 86 119 165)(53 164 120 85)(54 84 101 163)(55 162 102 83)(56 82 103 161)(57 180 104 81)(58 100 105 179)(59 178 106 99)(60 98 107 177)(121 187 231 141)(122 160 232 186)(123 185 233 159)(124 158 234 184)(125 183 235 157)(126 156 236 182)(127 181 237 155)(128 154 238 200)(129 199 239 153)(130 152 240 198)(131 197 221 151)(132 150 222 196)(133 195 223 149)(134 148 224 194)(135 193 225 147)(136 146 226 192)(137 191 227 145)(138 144 228 190)(139 189 229 143)(140 142 230 188)
G:=sub<Sym(240)| (1,164,133,28,85,223)(2,165,134,29,86,224)(3,166,135,30,87,225)(4,167,136,31,88,226)(5,168,137,32,89,227)(6,169,138,33,90,228)(7,170,139,34,91,229)(8,171,140,35,92,230)(9,172,121,36,93,231)(10,173,122,37,94,232)(11,174,123,38,95,233)(12,175,124,39,96,234)(13,176,125,40,97,235)(14,177,126,21,98,236)(15,178,127,22,99,237)(16,179,128,23,100,238)(17,180,129,24,81,239)(18,161,130,25,82,240)(19,162,131,26,83,221)(20,163,132,27,84,222)(41,157,215,108,183,62)(42,158,216,109,184,63)(43,159,217,110,185,64)(44,160,218,111,186,65)(45,141,219,112,187,66)(46,142,220,113,188,67)(47,143,201,114,189,68)(48,144,202,115,190,69)(49,145,203,116,191,70)(50,146,204,117,192,71)(51,147,205,118,193,72)(52,148,206,119,194,73)(53,149,207,120,195,74)(54,150,208,101,196,75)(55,151,209,102,197,76)(56,152,210,103,198,77)(57,153,211,104,199,78)(58,154,212,105,200,79)(59,155,213,106,181,80)(60,156,214,107,182,61), (1,223)(2,134)(3,225)(4,136)(5,227)(6,138)(7,229)(8,140)(9,231)(10,122)(11,233)(12,124)(13,235)(14,126)(15,237)(16,128)(17,239)(18,130)(19,221)(20,132)(21,236)(22,127)(23,238)(24,129)(25,240)(26,131)(27,222)(28,133)(29,224)(30,135)(31,226)(32,137)(33,228)(34,139)(35,230)(36,121)(37,232)(38,123)(39,234)(40,125)(41,108)(43,110)(45,112)(47,114)(49,116)(51,118)(53,120)(55,102)(57,104)(59,106)(61,156)(62,183)(63,158)(64,185)(65,160)(66,187)(67,142)(68,189)(69,144)(70,191)(71,146)(72,193)(73,148)(74,195)(75,150)(76,197)(77,152)(78,199)(79,154)(80,181)(81,180)(83,162)(85,164)(87,166)(89,168)(91,170)(93,172)(95,174)(97,176)(99,178)(141,219)(143,201)(145,203)(147,205)(149,207)(151,209)(153,211)(155,213)(157,215)(159,217)(182,214)(184,216)(186,218)(188,220)(190,202)(192,204)(194,206)(196,208)(198,210)(200,212), (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), (1,207,28,74)(2,73,29,206)(3,205,30,72)(4,71,31,204)(5,203,32,70)(6,69,33,202)(7,201,34,68)(8,67,35,220)(9,219,36,66)(10,65,37,218)(11,217,38,64)(12,63,39,216)(13,215,40,62)(14,61,21,214)(15,213,22,80)(16,79,23,212)(17,211,24,78)(18,77,25,210)(19,209,26,76)(20,75,27,208)(41,176,108,97)(42,96,109,175)(43,174,110,95)(44,94,111,173)(45,172,112,93)(46,92,113,171)(47,170,114,91)(48,90,115,169)(49,168,116,89)(50,88,117,167)(51,166,118,87)(52,86,119,165)(53,164,120,85)(54,84,101,163)(55,162,102,83)(56,82,103,161)(57,180,104,81)(58,100,105,179)(59,178,106,99)(60,98,107,177)(121,187,231,141)(122,160,232,186)(123,185,233,159)(124,158,234,184)(125,183,235,157)(126,156,236,182)(127,181,237,155)(128,154,238,200)(129,199,239,153)(130,152,240,198)(131,197,221,151)(132,150,222,196)(133,195,223,149)(134,148,224,194)(135,193,225,147)(136,146,226,192)(137,191,227,145)(138,144,228,190)(139,189,229,143)(140,142,230,188)>;
G:=Group( (1,164,133,28,85,223)(2,165,134,29,86,224)(3,166,135,30,87,225)(4,167,136,31,88,226)(5,168,137,32,89,227)(6,169,138,33,90,228)(7,170,139,34,91,229)(8,171,140,35,92,230)(9,172,121,36,93,231)(10,173,122,37,94,232)(11,174,123,38,95,233)(12,175,124,39,96,234)(13,176,125,40,97,235)(14,177,126,21,98,236)(15,178,127,22,99,237)(16,179,128,23,100,238)(17,180,129,24,81,239)(18,161,130,25,82,240)(19,162,131,26,83,221)(20,163,132,27,84,222)(41,157,215,108,183,62)(42,158,216,109,184,63)(43,159,217,110,185,64)(44,160,218,111,186,65)(45,141,219,112,187,66)(46,142,220,113,188,67)(47,143,201,114,189,68)(48,144,202,115,190,69)(49,145,203,116,191,70)(50,146,204,117,192,71)(51,147,205,118,193,72)(52,148,206,119,194,73)(53,149,207,120,195,74)(54,150,208,101,196,75)(55,151,209,102,197,76)(56,152,210,103,198,77)(57,153,211,104,199,78)(58,154,212,105,200,79)(59,155,213,106,181,80)(60,156,214,107,182,61), (1,223)(2,134)(3,225)(4,136)(5,227)(6,138)(7,229)(8,140)(9,231)(10,122)(11,233)(12,124)(13,235)(14,126)(15,237)(16,128)(17,239)(18,130)(19,221)(20,132)(21,236)(22,127)(23,238)(24,129)(25,240)(26,131)(27,222)(28,133)(29,224)(30,135)(31,226)(32,137)(33,228)(34,139)(35,230)(36,121)(37,232)(38,123)(39,234)(40,125)(41,108)(43,110)(45,112)(47,114)(49,116)(51,118)(53,120)(55,102)(57,104)(59,106)(61,156)(62,183)(63,158)(64,185)(65,160)(66,187)(67,142)(68,189)(69,144)(70,191)(71,146)(72,193)(73,148)(74,195)(75,150)(76,197)(77,152)(78,199)(79,154)(80,181)(81,180)(83,162)(85,164)(87,166)(89,168)(91,170)(93,172)(95,174)(97,176)(99,178)(141,219)(143,201)(145,203)(147,205)(149,207)(151,209)(153,211)(155,213)(157,215)(159,217)(182,214)(184,216)(186,218)(188,220)(190,202)(192,204)(194,206)(196,208)(198,210)(200,212), (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), (1,207,28,74)(2,73,29,206)(3,205,30,72)(4,71,31,204)(5,203,32,70)(6,69,33,202)(7,201,34,68)(8,67,35,220)(9,219,36,66)(10,65,37,218)(11,217,38,64)(12,63,39,216)(13,215,40,62)(14,61,21,214)(15,213,22,80)(16,79,23,212)(17,211,24,78)(18,77,25,210)(19,209,26,76)(20,75,27,208)(41,176,108,97)(42,96,109,175)(43,174,110,95)(44,94,111,173)(45,172,112,93)(46,92,113,171)(47,170,114,91)(48,90,115,169)(49,168,116,89)(50,88,117,167)(51,166,118,87)(52,86,119,165)(53,164,120,85)(54,84,101,163)(55,162,102,83)(56,82,103,161)(57,180,104,81)(58,100,105,179)(59,178,106,99)(60,98,107,177)(121,187,231,141)(122,160,232,186)(123,185,233,159)(124,158,234,184)(125,183,235,157)(126,156,236,182)(127,181,237,155)(128,154,238,200)(129,199,239,153)(130,152,240,198)(131,197,221,151)(132,150,222,196)(133,195,223,149)(134,148,224,194)(135,193,225,147)(136,146,226,192)(137,191,227,145)(138,144,228,190)(139,189,229,143)(140,142,230,188) );
G=PermutationGroup([[(1,164,133,28,85,223),(2,165,134,29,86,224),(3,166,135,30,87,225),(4,167,136,31,88,226),(5,168,137,32,89,227),(6,169,138,33,90,228),(7,170,139,34,91,229),(8,171,140,35,92,230),(9,172,121,36,93,231),(10,173,122,37,94,232),(11,174,123,38,95,233),(12,175,124,39,96,234),(13,176,125,40,97,235),(14,177,126,21,98,236),(15,178,127,22,99,237),(16,179,128,23,100,238),(17,180,129,24,81,239),(18,161,130,25,82,240),(19,162,131,26,83,221),(20,163,132,27,84,222),(41,157,215,108,183,62),(42,158,216,109,184,63),(43,159,217,110,185,64),(44,160,218,111,186,65),(45,141,219,112,187,66),(46,142,220,113,188,67),(47,143,201,114,189,68),(48,144,202,115,190,69),(49,145,203,116,191,70),(50,146,204,117,192,71),(51,147,205,118,193,72),(52,148,206,119,194,73),(53,149,207,120,195,74),(54,150,208,101,196,75),(55,151,209,102,197,76),(56,152,210,103,198,77),(57,153,211,104,199,78),(58,154,212,105,200,79),(59,155,213,106,181,80),(60,156,214,107,182,61)], [(1,223),(2,134),(3,225),(4,136),(5,227),(6,138),(7,229),(8,140),(9,231),(10,122),(11,233),(12,124),(13,235),(14,126),(15,237),(16,128),(17,239),(18,130),(19,221),(20,132),(21,236),(22,127),(23,238),(24,129),(25,240),(26,131),(27,222),(28,133),(29,224),(30,135),(31,226),(32,137),(33,228),(34,139),(35,230),(36,121),(37,232),(38,123),(39,234),(40,125),(41,108),(43,110),(45,112),(47,114),(49,116),(51,118),(53,120),(55,102),(57,104),(59,106),(61,156),(62,183),(63,158),(64,185),(65,160),(66,187),(67,142),(68,189),(69,144),(70,191),(71,146),(72,193),(73,148),(74,195),(75,150),(76,197),(77,152),(78,199),(79,154),(80,181),(81,180),(83,162),(85,164),(87,166),(89,168),(91,170),(93,172),(95,174),(97,176),(99,178),(141,219),(143,201),(145,203),(147,205),(149,207),(151,209),(153,211),(155,213),(157,215),(159,217),(182,214),(184,216),(186,218),(188,220),(190,202),(192,204),(194,206),(196,208),(198,210),(200,212)], [(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)], [(1,207,28,74),(2,73,29,206),(3,205,30,72),(4,71,31,204),(5,203,32,70),(6,69,33,202),(7,201,34,68),(8,67,35,220),(9,219,36,66),(10,65,37,218),(11,217,38,64),(12,63,39,216),(13,215,40,62),(14,61,21,214),(15,213,22,80),(16,79,23,212),(17,211,24,78),(18,77,25,210),(19,209,26,76),(20,75,27,208),(41,176,108,97),(42,96,109,175),(43,174,110,95),(44,94,111,173),(45,172,112,93),(46,92,113,171),(47,170,114,91),(48,90,115,169),(49,168,116,89),(50,88,117,167),(51,166,118,87),(52,86,119,165),(53,164,120,85),(54,84,101,163),(55,162,102,83),(56,82,103,161),(57,180,104,81),(58,100,105,179),(59,178,106,99),(60,98,107,177),(121,187,231,141),(122,160,232,186),(123,185,233,159),(124,158,234,184),(125,183,235,157),(126,156,236,182),(127,181,237,155),(128,154,238,200),(129,199,239,153),(130,152,240,198),(131,197,221,151),(132,150,222,196),(133,195,223,149),(134,148,224,194),(135,193,225,147),(136,146,226,192),(137,191,227,145),(138,144,228,190),(139,189,229,143),(140,142,230,188)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 20 | 2 | 4 | 10 | 10 | 12 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D20 | C4○D12 | S3×D4 | D4⋊2S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊5D5 | S3×D20 | C30.C23 |
| kernel | D6.9D20 | D10⋊Dic3 | C6.Dic10 | C3×D10⋊C4 | C5×D6⋊C4 | C60⋊5C4 | C2×S3×Dic5 | C2×C15⋊D4 | D10⋊C4 | S3×C10 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D6.9D20 ►in GL4(𝔽61) generated by
| 60 | 19 | 0 | 0 |
| 48 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 59 | 19 | 0 | 0 |
| 48 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 53 | 20 | 0 | 0 |
| 12 | 8 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 3 | 28 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 41 | 14 |
| 0 | 0 | 2 | 20 |
G:=sub<GL(4,GF(61))| [60,48,0,0,19,2,0,0,0,0,1,0,0,0,0,1],[59,48,0,0,19,2,0,0,0,0,1,0,0,0,0,1],[53,12,0,0,20,8,0,0,0,0,24,3,0,0,0,28],[11,0,0,0,0,11,0,0,0,0,41,2,0,0,14,20] >;
D6.9D20 in GAP, Magma, Sage, TeX
D_6._9D_{20} % in TeX
G:=Group("D6.9D20"); // GroupNames label
G:=SmallGroup(480,533);
// by ID
G=gap.SmallGroup(480,533);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations