metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.39D6, C30.5C24, D12.39D10, C15⋊22- (1+4), C60.111C23, Dic6.42D10, Dic10.42D6, Dic15.4C23, Dic30.56C22, C4○D20⋊6S3, C4○D12⋊5D5, C5⋊2(Q8○D12), C5⋊D4.2D6, (C4×D5).14D6, C3⋊D4.2D10, C15⋊D4.C22, C6.5(C23×D5), C15⋊Q8.2C22, (D5×Dic6)⋊12C2, (C4×S3).13D10, (C2×C20).164D6, C10.5(S3×C23), D20⋊5S3⋊11C2, D12⋊5D5⋊11C2, (C6×D5).4C23, D6.1(C22×D5), (S3×Dic10)⋊11C2, (C2×Dic30)⋊20C2, C30.C23⋊1C2, (C2×C12).162D10, (S3×C10).1C23, (C2×C60).97C22, D10.4(C22×S3), C3⋊2(D4.10D10), (S3×C20).28C22, (C2×C30).224C23, C20.124(C22×S3), (D5×C12).29C22, (C5×D12).39C22, (C3×D20).39C22, C12.123(C22×D5), Dic3.4(C22×D5), (C3×Dic5).2C23, Dic5.2(C22×S3), (C5×Dic3).4C23, (S3×Dic5).1C22, (D5×Dic3).2C22, (C5×Dic6).42C22, (C3×Dic10).41C22, (C2×Dic15).149C22, C4.85(C2×S3×D5), (C3×C4○D20)⋊7C2, (C5×C4○D12)⋊7C2, C2.9(C22×S3×D5), (C2×C4).65(S3×D5), C22.17(C2×S3×D5), (C2×C6).9(C22×D5), (C2×C10).8(C22×S3), (C3×C5⋊D4).3C22, (C5×C3⋊D4).2C22, SmallGroup(480,1077)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1292 in 292 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], C5, S3 [×2], C6, C6 [×3], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×2], C2×C6, C2×C6 [×2], C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], Dic6, Dic6 [×8], C4×S3 [×2], C4×S3 [×4], D12, C2×Dic3 [×6], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C5×S3 [×2], C3×D5 [×2], C30, C30, 2- (1+4), Dic10, Dic10 [×8], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C2×Dic6 [×3], C4○D12, C4○D12 [×2], D4⋊2S3 [×6], S3×Q8 [×2], C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C60 [×2], C6×D5 [×2], S3×C10 [×2], C2×C30, C2×Dic10 [×3], C4○D20, C4○D20 [×2], D4⋊2D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8○D12, D5×Dic3 [×4], S3×Dic5 [×4], C15⋊D4 [×4], C15⋊Q8 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30 [×4], C2×Dic15 [×2], C2×C60, D4.10D10, D5×Dic6 [×2], D20⋊5S3 [×2], S3×Dic10 [×2], D12⋊5D5 [×2], C30.C23 [×4], C3×C4○D20, C5×C4○D12, C2×Dic30, D20.39D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, D20.39D6
Generators and relations
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a10c5 >
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)(161 173)(162 172)(163 171)(164 170)(165 169)(166 168)(174 180)(175 179)(176 178)(181 183)(184 200)(185 199)(186 198)(187 197)(188 196)(189 195)(190 194)(191 193)(201 207)(202 206)(203 205)(208 220)(209 219)(210 218)(211 217)(212 216)(213 215)(221 231)(222 230)(223 229)(224 228)(225 227)(232 240)(233 239)(234 238)(235 237)
(1 86 110 137 48 77 11 96 120 127 58 67)(2 87 111 138 49 78 12 97 101 128 59 68)(3 88 112 139 50 79 13 98 102 129 60 69)(4 89 113 140 51 80 14 99 103 130 41 70)(5 90 114 121 52 61 15 100 104 131 42 71)(6 91 115 122 53 62 16 81 105 132 43 72)(7 92 116 123 54 63 17 82 106 133 44 73)(8 93 117 124 55 64 18 83 107 134 45 74)(9 94 118 125 56 65 19 84 108 135 46 75)(10 95 119 126 57 66 20 85 109 136 47 76)(21 167 158 192 236 204 31 177 148 182 226 214)(22 168 159 193 237 205 32 178 149 183 227 215)(23 169 160 194 238 206 33 179 150 184 228 216)(24 170 141 195 239 207 34 180 151 185 229 217)(25 171 142 196 240 208 35 161 152 186 230 218)(26 172 143 197 221 209 36 162 153 187 231 219)(27 173 144 198 222 210 37 163 154 188 232 220)(28 174 145 199 223 211 38 164 155 189 233 201)(29 175 146 200 224 212 39 165 156 190 234 202)(30 176 147 181 225 213 40 166 157 191 235 203)
(1 229 11 239)(2 230 12 240)(3 231 13 221)(4 232 14 222)(5 233 15 223)(6 234 16 224)(7 235 17 225)(8 236 18 226)(9 237 19 227)(10 238 20 228)(21 45 31 55)(22 46 32 56)(23 47 33 57)(24 48 34 58)(25 49 35 59)(26 50 36 60)(27 51 37 41)(28 52 38 42)(29 53 39 43)(30 54 40 44)(61 211 71 201)(62 212 72 202)(63 213 73 203)(64 214 74 204)(65 215 75 205)(66 216 76 206)(67 217 77 207)(68 218 78 208)(69 219 79 209)(70 220 80 210)(81 200 91 190)(82 181 92 191)(83 182 93 192)(84 183 94 193)(85 184 95 194)(86 185 96 195)(87 186 97 196)(88 187 98 197)(89 188 99 198)(90 189 100 199)(101 142 111 152)(102 143 112 153)(103 144 113 154)(104 145 114 155)(105 146 115 156)(106 147 116 157)(107 148 117 158)(108 149 118 159)(109 150 119 160)(110 151 120 141)(121 164 131 174)(122 165 132 175)(123 166 133 176)(124 167 134 177)(125 168 135 178)(126 169 136 179)(127 170 137 180)(128 171 138 161)(129 172 139 162)(130 173 140 163)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)(161,173)(162,172)(163,171)(164,170)(165,169)(166,168)(174,180)(175,179)(176,178)(181,183)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(201,207)(202,206)(203,205)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215)(221,231)(222,230)(223,229)(224,228)(225,227)(232,240)(233,239)(234,238)(235,237), (1,86,110,137,48,77,11,96,120,127,58,67)(2,87,111,138,49,78,12,97,101,128,59,68)(3,88,112,139,50,79,13,98,102,129,60,69)(4,89,113,140,51,80,14,99,103,130,41,70)(5,90,114,121,52,61,15,100,104,131,42,71)(6,91,115,122,53,62,16,81,105,132,43,72)(7,92,116,123,54,63,17,82,106,133,44,73)(8,93,117,124,55,64,18,83,107,134,45,74)(9,94,118,125,56,65,19,84,108,135,46,75)(10,95,119,126,57,66,20,85,109,136,47,76)(21,167,158,192,236,204,31,177,148,182,226,214)(22,168,159,193,237,205,32,178,149,183,227,215)(23,169,160,194,238,206,33,179,150,184,228,216)(24,170,141,195,239,207,34,180,151,185,229,217)(25,171,142,196,240,208,35,161,152,186,230,218)(26,172,143,197,221,209,36,162,153,187,231,219)(27,173,144,198,222,210,37,163,154,188,232,220)(28,174,145,199,223,211,38,164,155,189,233,201)(29,175,146,200,224,212,39,165,156,190,234,202)(30,176,147,181,225,213,40,166,157,191,235,203), (1,229,11,239)(2,230,12,240)(3,231,13,221)(4,232,14,222)(5,233,15,223)(6,234,16,224)(7,235,17,225)(8,236,18,226)(9,237,19,227)(10,238,20,228)(21,45,31,55)(22,46,32,56)(23,47,33,57)(24,48,34,58)(25,49,35,59)(26,50,36,60)(27,51,37,41)(28,52,38,42)(29,53,39,43)(30,54,40,44)(61,211,71,201)(62,212,72,202)(63,213,73,203)(64,214,74,204)(65,215,75,205)(66,216,76,206)(67,217,77,207)(68,218,78,208)(69,219,79,209)(70,220,80,210)(81,200,91,190)(82,181,92,191)(83,182,93,192)(84,183,94,193)(85,184,95,194)(86,185,96,195)(87,186,97,196)(88,187,98,197)(89,188,99,198)(90,189,100,199)(101,142,111,152)(102,143,112,153)(103,144,113,154)(104,145,114,155)(105,146,115,156)(106,147,116,157)(107,148,117,158)(108,149,118,159)(109,150,119,160)(110,151,120,141)(121,164,131,174)(122,165,132,175)(123,166,133,176)(124,167,134,177)(125,168,135,178)(126,169,136,179)(127,170,137,180)(128,171,138,161)(129,172,139,162)(130,173,140,163)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,85)(82,84)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)(161,173)(162,172)(163,171)(164,170)(165,169)(166,168)(174,180)(175,179)(176,178)(181,183)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(201,207)(202,206)(203,205)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215)(221,231)(222,230)(223,229)(224,228)(225,227)(232,240)(233,239)(234,238)(235,237), (1,86,110,137,48,77,11,96,120,127,58,67)(2,87,111,138,49,78,12,97,101,128,59,68)(3,88,112,139,50,79,13,98,102,129,60,69)(4,89,113,140,51,80,14,99,103,130,41,70)(5,90,114,121,52,61,15,100,104,131,42,71)(6,91,115,122,53,62,16,81,105,132,43,72)(7,92,116,123,54,63,17,82,106,133,44,73)(8,93,117,124,55,64,18,83,107,134,45,74)(9,94,118,125,56,65,19,84,108,135,46,75)(10,95,119,126,57,66,20,85,109,136,47,76)(21,167,158,192,236,204,31,177,148,182,226,214)(22,168,159,193,237,205,32,178,149,183,227,215)(23,169,160,194,238,206,33,179,150,184,228,216)(24,170,141,195,239,207,34,180,151,185,229,217)(25,171,142,196,240,208,35,161,152,186,230,218)(26,172,143,197,221,209,36,162,153,187,231,219)(27,173,144,198,222,210,37,163,154,188,232,220)(28,174,145,199,223,211,38,164,155,189,233,201)(29,175,146,200,224,212,39,165,156,190,234,202)(30,176,147,181,225,213,40,166,157,191,235,203), (1,229,11,239)(2,230,12,240)(3,231,13,221)(4,232,14,222)(5,233,15,223)(6,234,16,224)(7,235,17,225)(8,236,18,226)(9,237,19,227)(10,238,20,228)(21,45,31,55)(22,46,32,56)(23,47,33,57)(24,48,34,58)(25,49,35,59)(26,50,36,60)(27,51,37,41)(28,52,38,42)(29,53,39,43)(30,54,40,44)(61,211,71,201)(62,212,72,202)(63,213,73,203)(64,214,74,204)(65,215,75,205)(66,216,76,206)(67,217,77,207)(68,218,78,208)(69,219,79,209)(70,220,80,210)(81,200,91,190)(82,181,92,191)(83,182,93,192)(84,183,94,193)(85,184,95,194)(86,185,96,195)(87,186,97,196)(88,187,98,197)(89,188,99,198)(90,189,100,199)(101,142,111,152)(102,143,112,153)(103,144,113,154)(104,145,114,155)(105,146,115,156)(106,147,116,157)(107,148,117,158)(108,149,118,159)(109,150,119,160)(110,151,120,141)(121,164,131,174)(122,165,132,175)(123,166,133,176)(124,167,134,177)(125,168,135,178)(126,169,136,179)(127,170,137,180)(128,171,138,161)(129,172,139,162)(130,173,140,163) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,85),(82,84),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(114,120),(115,119),(116,118),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159),(161,173),(162,172),(163,171),(164,170),(165,169),(166,168),(174,180),(175,179),(176,178),(181,183),(184,200),(185,199),(186,198),(187,197),(188,196),(189,195),(190,194),(191,193),(201,207),(202,206),(203,205),(208,220),(209,219),(210,218),(211,217),(212,216),(213,215),(221,231),(222,230),(223,229),(224,228),(225,227),(232,240),(233,239),(234,238),(235,237)], [(1,86,110,137,48,77,11,96,120,127,58,67),(2,87,111,138,49,78,12,97,101,128,59,68),(3,88,112,139,50,79,13,98,102,129,60,69),(4,89,113,140,51,80,14,99,103,130,41,70),(5,90,114,121,52,61,15,100,104,131,42,71),(6,91,115,122,53,62,16,81,105,132,43,72),(7,92,116,123,54,63,17,82,106,133,44,73),(8,93,117,124,55,64,18,83,107,134,45,74),(9,94,118,125,56,65,19,84,108,135,46,75),(10,95,119,126,57,66,20,85,109,136,47,76),(21,167,158,192,236,204,31,177,148,182,226,214),(22,168,159,193,237,205,32,178,149,183,227,215),(23,169,160,194,238,206,33,179,150,184,228,216),(24,170,141,195,239,207,34,180,151,185,229,217),(25,171,142,196,240,208,35,161,152,186,230,218),(26,172,143,197,221,209,36,162,153,187,231,219),(27,173,144,198,222,210,37,163,154,188,232,220),(28,174,145,199,223,211,38,164,155,189,233,201),(29,175,146,200,224,212,39,165,156,190,234,202),(30,176,147,181,225,213,40,166,157,191,235,203)], [(1,229,11,239),(2,230,12,240),(3,231,13,221),(4,232,14,222),(5,233,15,223),(6,234,16,224),(7,235,17,225),(8,236,18,226),(9,237,19,227),(10,238,20,228),(21,45,31,55),(22,46,32,56),(23,47,33,57),(24,48,34,58),(25,49,35,59),(26,50,36,60),(27,51,37,41),(28,52,38,42),(29,53,39,43),(30,54,40,44),(61,211,71,201),(62,212,72,202),(63,213,73,203),(64,214,74,204),(65,215,75,205),(66,216,76,206),(67,217,77,207),(68,218,78,208),(69,219,79,209),(70,220,80,210),(81,200,91,190),(82,181,92,191),(83,182,93,192),(84,183,94,193),(85,184,95,194),(86,185,96,195),(87,186,97,196),(88,187,98,197),(89,188,99,198),(90,189,100,199),(101,142,111,152),(102,143,112,153),(103,144,113,154),(104,145,114,155),(105,146,115,156),(106,147,116,157),(107,148,117,158),(108,149,118,159),(109,150,119,160),(110,151,120,141),(121,164,131,174),(122,165,132,175),(123,166,133,176),(124,167,134,177),(125,168,135,178),(126,169,136,179),(127,170,137,180),(128,171,138,161),(129,172,139,162),(130,173,140,163)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 7 | 32 |
| 0 | 0 | 29 | 2 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 23 | 0 | 38 | 0 |
| 0 | 23 | 0 | 38 |
| 23 | 0 | 46 | 0 |
| 0 | 23 | 0 | 46 |
| 2 | 0 | 23 | 0 |
| 0 | 2 | 0 | 23 |
| 21 | 0 | 59 | 0 |
| 0 | 21 | 0 | 59 |
G:=sub<GL(4,GF(61))| [7,29,0,0,32,2,0,0,0,0,7,29,0,0,32,2],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0],[23,0,23,0,0,23,0,23,38,0,46,0,0,38,0,46],[2,0,21,0,0,2,0,21,23,0,59,0,0,23,0,59] >;
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2- (1+4) | S3×D5 | Q8○D12 | C2×S3×D5 | C2×S3×D5 | D4.10D10 | D20.39D6 |
| kernel | D20.39D6 | D5×Dic6 | D20⋊5S3 | S3×Dic10 | D12⋊5D5 | C30.C23 | C3×C4○D20 | C5×C4○D12 | C2×Dic30 | C4○D20 | C4○D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
D_{20}._{39}D_6 % in TeX
G:=Group("D20.39D6"); // GroupNames label
G:=SmallGroup(480,1077);
// by ID
G=gap.SmallGroup(480,1077);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^10*c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀