metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.29D6, C30.32C24, C60.56C23, C15⋊52- (1+4), Dic6.30D10, D30.39C23, Dic15.19C23, Dic30.20C22, (S3×Q8)⋊4D5, (Q8×D15)⋊4C2, C5⋊4(Q8○D12), Q8⋊2D5⋊7S3, (D5×Dic6)⋊6C2, (C4×D5).16D6, (C5×Q8).41D6, Q8.26(S3×D5), D20⋊5S3⋊6C2, D20⋊S3⋊6C2, C15⋊Q8.5C22, (C4×S3).16D10, (C3×Q8).24D10, C6.32(C23×D5), D6.D10⋊5C2, C20.56(C22×S3), C10.32(S3×C23), (C6×D5).14C23, D6.28(C22×D5), C12.56(C22×D5), C15⋊D4.3C22, C5⋊D12.5C22, C3⋊D20.3C22, (S3×C20).19C22, (S3×C10).31C23, C3⋊3(Q8.10D10), (C3×D20).19C22, (C4×D15).19C22, (D5×C12).19C22, D10.17(C22×S3), (Q8×C15).19C22, (D5×Dic3).4C22, (C5×Dic3).18C23, (C5×Dic6).20C22, (C3×Dic5).47C23, Dic5.44(C22×S3), Dic3.17(C22×D5), (C5×S3×Q8)⋊4C2, C4.56(C2×S3×D5), C2.35(C22×S3×D5), (C3×Q8⋊2D5)⋊4C2, SmallGroup(480,1104)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1340 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C4 [×7], C22 [×5], C5, S3 [×2], C6, C6 [×3], C2×C4 [×15], D4 [×10], Q8, Q8 [×9], D5 [×4], C10, C10, Dic3 [×3], Dic3 [×3], C12 [×3], C12, D6, D6, C2×C6 [×3], C15, C2×Q8 [×5], C4○D4 [×10], Dic5, Dic5 [×3], C20 [×3], C20 [×3], D10 [×3], D10, C2×C10, Dic6 [×3], Dic6 [×6], C4×S3 [×3], C4×S3 [×3], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C5×S3, C3×D5 [×3], D15, C30, 2- (1+4), Dic10 [×6], C4×D5 [×3], C4×D5 [×9], D20 [×3], D20 [×3], C5⋊D4 [×4], C2×C20 [×3], C5×Q8, C5×Q8 [×3], C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3 [×6], S3×Q8, S3×Q8, C3×C4○D4, C5×Dic3 [×3], C3×Dic5, Dic15 [×3], C60 [×3], C6×D5 [×3], S3×C10, D30, C4○D20 [×6], Q8×D5 [×4], Q8⋊2D5, Q8⋊2D5 [×3], Q8×C10, Q8○D12, D5×Dic3 [×6], C15⋊D4 [×3], C3⋊D20 [×3], C5⋊D12, C15⋊Q8 [×3], D5×C12 [×3], C3×D20 [×3], C5×Dic6 [×3], S3×C20 [×3], Dic30 [×3], C4×D15 [×3], Q8×C15, Q8.10D10, D5×Dic6 [×3], D20⋊5S3 [×3], D20⋊S3 [×3], D6.D10 [×3], C3×Q8⋊2D5, C5×S3×Q8, Q8×D15, D20.29D6
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], Q8.10D10, C22×S3×D5, D20.29D6
Generators and relations
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, 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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 60)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 100)(61 200)(62 199)(63 198)(64 197)(65 196)(66 195)(67 194)(68 193)(69 192)(70 191)(71 190)(72 189)(73 188)(74 187)(75 186)(76 185)(77 184)(78 183)(79 182)(80 181)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 140)(120 139)(141 239)(142 238)(143 237)(144 236)(145 235)(146 234)(147 233)(148 232)(149 231)(150 230)(151 229)(152 228)(153 227)(154 226)(155 225)(156 224)(157 223)(158 222)(159 221)(160 240)(161 213)(162 212)(163 211)(164 210)(165 209)(166 208)(167 207)(168 206)(169 205)(170 204)(171 203)(172 202)(173 201)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)
(1 79 108 235 92 168 11 69 118 225 82 178)(2 70 109 226 93 179 12 80 119 236 83 169)(3 61 110 237 94 170 13 71 120 227 84 180)(4 72 111 228 95 161 14 62 101 238 85 171)(5 63 112 239 96 172 15 73 102 229 86 162)(6 74 113 230 97 163 16 64 103 240 87 173)(7 65 114 221 98 174 17 75 104 231 88 164)(8 76 115 232 99 165 18 66 105 222 89 175)(9 67 116 223 100 176 19 77 106 233 90 166)(10 78 117 234 81 167 20 68 107 224 91 177)(21 219 42 185 134 148 31 209 52 195 124 158)(22 210 43 196 135 159 32 220 53 186 125 149)(23 201 44 187 136 150 33 211 54 197 126 160)(24 212 45 198 137 141 34 202 55 188 127 151)(25 203 46 189 138 152 35 213 56 199 128 142)(26 214 47 200 139 143 36 204 57 190 129 153)(27 205 48 191 140 154 37 215 58 181 130 144)(28 216 49 182 121 145 38 206 59 192 131 155)(29 207 50 193 122 156 39 217 60 183 132 146)(30 218 51 184 123 147 40 208 41 194 133 157)
(1 202 11 212)(2 203 12 213)(3 204 13 214)(4 205 14 215)(5 206 15 216)(6 207 16 217)(7 208 17 218)(8 209 18 219)(9 210 19 220)(10 211 20 201)(21 76 31 66)(22 77 32 67)(23 78 33 68)(24 79 34 69)(25 80 35 70)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)(41 174 51 164)(42 175 52 165)(43 176 53 166)(44 177 54 167)(45 178 55 168)(46 179 56 169)(47 180 57 170)(48 161 58 171)(49 162 59 172)(50 163 60 173)(81 187 91 197)(82 188 92 198)(83 189 93 199)(84 190 94 200)(85 191 95 181)(86 192 96 182)(87 193 97 183)(88 194 98 184)(89 195 99 185)(90 196 100 186)(101 154 111 144)(102 155 112 145)(103 156 113 146)(104 157 114 147)(105 158 115 148)(106 159 116 149)(107 160 117 150)(108 141 118 151)(109 142 119 152)(110 143 120 153)(121 229 131 239)(122 230 132 240)(123 231 133 221)(124 232 134 222)(125 233 135 223)(126 234 136 224)(127 235 137 225)(128 236 138 226)(129 237 139 227)(130 238 140 228)
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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,60)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,100)(61,200)(62,199)(63,198)(64,197)(65,196)(66,195)(67,194)(68,193)(69,192)(70,191)(71,190)(72,189)(73,188)(74,187)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,140)(120,139)(141,239)(142,238)(143,237)(144,236)(145,235)(146,234)(147,233)(148,232)(149,231)(150,230)(151,229)(152,228)(153,227)(154,226)(155,225)(156,224)(157,223)(158,222)(159,221)(160,240)(161,213)(162,212)(163,211)(164,210)(165,209)(166,208)(167,207)(168,206)(169,205)(170,204)(171,203)(172,202)(173,201)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214), (1,79,108,235,92,168,11,69,118,225,82,178)(2,70,109,226,93,179,12,80,119,236,83,169)(3,61,110,237,94,170,13,71,120,227,84,180)(4,72,111,228,95,161,14,62,101,238,85,171)(5,63,112,239,96,172,15,73,102,229,86,162)(6,74,113,230,97,163,16,64,103,240,87,173)(7,65,114,221,98,174,17,75,104,231,88,164)(8,76,115,232,99,165,18,66,105,222,89,175)(9,67,116,223,100,176,19,77,106,233,90,166)(10,78,117,234,81,167,20,68,107,224,91,177)(21,219,42,185,134,148,31,209,52,195,124,158)(22,210,43,196,135,159,32,220,53,186,125,149)(23,201,44,187,136,150,33,211,54,197,126,160)(24,212,45,198,137,141,34,202,55,188,127,151)(25,203,46,189,138,152,35,213,56,199,128,142)(26,214,47,200,139,143,36,204,57,190,129,153)(27,205,48,191,140,154,37,215,58,181,130,144)(28,216,49,182,121,145,38,206,59,192,131,155)(29,207,50,193,122,156,39,217,60,183,132,146)(30,218,51,184,123,147,40,208,41,194,133,157), (1,202,11,212)(2,203,12,213)(3,204,13,214)(4,205,14,215)(5,206,15,216)(6,207,16,217)(7,208,17,218)(8,209,18,219)(9,210,19,220)(10,211,20,201)(21,76,31,66)(22,77,32,67)(23,78,33,68)(24,79,34,69)(25,80,35,70)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75)(41,174,51,164)(42,175,52,165)(43,176,53,166)(44,177,54,167)(45,178,55,168)(46,179,56,169)(47,180,57,170)(48,161,58,171)(49,162,59,172)(50,163,60,173)(81,187,91,197)(82,188,92,198)(83,189,93,199)(84,190,94,200)(85,191,95,181)(86,192,96,182)(87,193,97,183)(88,194,98,184)(89,195,99,185)(90,196,100,186)(101,154,111,144)(102,155,112,145)(103,156,113,146)(104,157,114,147)(105,158,115,148)(106,159,116,149)(107,160,117,150)(108,141,118,151)(109,142,119,152)(110,143,120,153)(121,229,131,239)(122,230,132,240)(123,231,133,221)(124,232,134,222)(125,233,135,223)(126,234,136,224)(127,235,137,225)(128,236,138,226)(129,237,139,227)(130,238,140,228)>;
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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,60)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,100)(61,200)(62,199)(63,198)(64,197)(65,196)(66,195)(67,194)(68,193)(69,192)(70,191)(71,190)(72,189)(73,188)(74,187)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,140)(120,139)(141,239)(142,238)(143,237)(144,236)(145,235)(146,234)(147,233)(148,232)(149,231)(150,230)(151,229)(152,228)(153,227)(154,226)(155,225)(156,224)(157,223)(158,222)(159,221)(160,240)(161,213)(162,212)(163,211)(164,210)(165,209)(166,208)(167,207)(168,206)(169,205)(170,204)(171,203)(172,202)(173,201)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214), (1,79,108,235,92,168,11,69,118,225,82,178)(2,70,109,226,93,179,12,80,119,236,83,169)(3,61,110,237,94,170,13,71,120,227,84,180)(4,72,111,228,95,161,14,62,101,238,85,171)(5,63,112,239,96,172,15,73,102,229,86,162)(6,74,113,230,97,163,16,64,103,240,87,173)(7,65,114,221,98,174,17,75,104,231,88,164)(8,76,115,232,99,165,18,66,105,222,89,175)(9,67,116,223,100,176,19,77,106,233,90,166)(10,78,117,234,81,167,20,68,107,224,91,177)(21,219,42,185,134,148,31,209,52,195,124,158)(22,210,43,196,135,159,32,220,53,186,125,149)(23,201,44,187,136,150,33,211,54,197,126,160)(24,212,45,198,137,141,34,202,55,188,127,151)(25,203,46,189,138,152,35,213,56,199,128,142)(26,214,47,200,139,143,36,204,57,190,129,153)(27,205,48,191,140,154,37,215,58,181,130,144)(28,216,49,182,121,145,38,206,59,192,131,155)(29,207,50,193,122,156,39,217,60,183,132,146)(30,218,51,184,123,147,40,208,41,194,133,157), (1,202,11,212)(2,203,12,213)(3,204,13,214)(4,205,14,215)(5,206,15,216)(6,207,16,217)(7,208,17,218)(8,209,18,219)(9,210,19,220)(10,211,20,201)(21,76,31,66)(22,77,32,67)(23,78,33,68)(24,79,34,69)(25,80,35,70)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75)(41,174,51,164)(42,175,52,165)(43,176,53,166)(44,177,54,167)(45,178,55,168)(46,179,56,169)(47,180,57,170)(48,161,58,171)(49,162,59,172)(50,163,60,173)(81,187,91,197)(82,188,92,198)(83,189,93,199)(84,190,94,200)(85,191,95,181)(86,192,96,182)(87,193,97,183)(88,194,98,184)(89,195,99,185)(90,196,100,186)(101,154,111,144)(102,155,112,145)(103,156,113,146)(104,157,114,147)(105,158,115,148)(106,159,116,149)(107,160,117,150)(108,141,118,151)(109,142,119,152)(110,143,120,153)(121,229,131,239)(122,230,132,240)(123,231,133,221)(124,232,134,222)(125,233,135,223)(126,234,136,224)(127,235,137,225)(128,236,138,226)(129,237,139,227)(130,238,140,228) );
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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,60),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,100),(61,200),(62,199),(63,198),(64,197),(65,196),(66,195),(67,194),(68,193),(69,192),(70,191),(71,190),(72,189),(73,188),(74,187),(75,186),(76,185),(77,184),(78,183),(79,182),(80,181),(101,138),(102,137),(103,136),(104,135),(105,134),(106,133),(107,132),(108,131),(109,130),(110,129),(111,128),(112,127),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,140),(120,139),(141,239),(142,238),(143,237),(144,236),(145,235),(146,234),(147,233),(148,232),(149,231),(150,230),(151,229),(152,228),(153,227),(154,226),(155,225),(156,224),(157,223),(158,222),(159,221),(160,240),(161,213),(162,212),(163,211),(164,210),(165,209),(166,208),(167,207),(168,206),(169,205),(170,204),(171,203),(172,202),(173,201),(174,220),(175,219),(176,218),(177,217),(178,216),(179,215),(180,214)], [(1,79,108,235,92,168,11,69,118,225,82,178),(2,70,109,226,93,179,12,80,119,236,83,169),(3,61,110,237,94,170,13,71,120,227,84,180),(4,72,111,228,95,161,14,62,101,238,85,171),(5,63,112,239,96,172,15,73,102,229,86,162),(6,74,113,230,97,163,16,64,103,240,87,173),(7,65,114,221,98,174,17,75,104,231,88,164),(8,76,115,232,99,165,18,66,105,222,89,175),(9,67,116,223,100,176,19,77,106,233,90,166),(10,78,117,234,81,167,20,68,107,224,91,177),(21,219,42,185,134,148,31,209,52,195,124,158),(22,210,43,196,135,159,32,220,53,186,125,149),(23,201,44,187,136,150,33,211,54,197,126,160),(24,212,45,198,137,141,34,202,55,188,127,151),(25,203,46,189,138,152,35,213,56,199,128,142),(26,214,47,200,139,143,36,204,57,190,129,153),(27,205,48,191,140,154,37,215,58,181,130,144),(28,216,49,182,121,145,38,206,59,192,131,155),(29,207,50,193,122,156,39,217,60,183,132,146),(30,218,51,184,123,147,40,208,41,194,133,157)], [(1,202,11,212),(2,203,12,213),(3,204,13,214),(4,205,14,215),(5,206,15,216),(6,207,16,217),(7,208,17,218),(8,209,18,219),(9,210,19,220),(10,211,20,201),(21,76,31,66),(22,77,32,67),(23,78,33,68),(24,79,34,69),(25,80,35,70),(26,61,36,71),(27,62,37,72),(28,63,38,73),(29,64,39,74),(30,65,40,75),(41,174,51,164),(42,175,52,165),(43,176,53,166),(44,177,54,167),(45,178,55,168),(46,179,56,169),(47,180,57,170),(48,161,58,171),(49,162,59,172),(50,163,60,173),(81,187,91,197),(82,188,92,198),(83,189,93,199),(84,190,94,200),(85,191,95,181),(86,192,96,182),(87,193,97,183),(88,194,98,184),(89,195,99,185),(90,196,100,186),(101,154,111,144),(102,155,112,145),(103,156,113,146),(104,157,114,147),(105,158,115,148),(106,159,116,149),(107,160,117,150),(108,141,118,151),(109,142,119,152),(110,143,120,153),(121,229,131,239),(122,230,132,240),(123,231,133,221),(124,232,134,222),(125,233,135,223),(126,234,136,224),(127,235,137,225),(128,236,138,226),(129,237,139,227),(130,238,140,228)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 30 | 38 | 13 |
| 0 | 0 | 0 | 0 | 53 | 31 | 23 | 25 |
| 0 | 0 | 0 | 0 | 51 | 0 | 0 | 31 |
| 0 | 0 | 0 | 0 | 48 | 51 | 30 | 52 |
| 35 | 2 | 9 | 15 | 0 | 0 | 0 | 0 |
| 48 | 26 | 46 | 52 | 0 | 0 | 0 | 0 |
| 52 | 46 | 26 | 48 | 0 | 0 | 0 | 0 |
| 15 | 9 | 2 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 30 | 13 | 15 |
| 0 | 0 | 0 | 0 | 28 | 14 | 2 | 59 |
| 0 | 0 | 0 | 0 | 48 | 55 | 17 | 31 |
| 0 | 0 | 0 | 0 | 48 | 54 | 30 | 44 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 | 0 | 25 |
| 0 | 0 | 0 | 0 | 0 | 47 | 36 | 36 |
| 0 | 0 | 0 | 0 | 3 | 3 | 14 | 0 |
| 0 | 0 | 0 | 0 | 58 | 0 | 0 | 14 |
| 2 | 35 | 48 | 26 | 0 | 0 | 0 | 0 |
| 26 | 48 | 35 | 2 | 0 | 0 | 0 | 0 |
| 46 | 52 | 59 | 26 | 0 | 0 | 0 | 0 |
| 9 | 15 | 35 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 39 | 25 | 42 |
| 0 | 0 | 0 | 0 | 14 | 7 | 17 | 44 |
| 0 | 0 | 0 | 0 | 43 | 56 | 15 | 22 |
| 0 | 0 | 0 | 0 | 43 | 48 | 39 | 46 |
G:=sub<GL(8,GF(61))| [0,1,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,0,39,53,51,48,0,0,0,0,30,31,0,51,0,0,0,0,38,23,0,30,0,0,0,0,13,25,31,52],[35,48,52,15,0,0,0,0,2,26,46,9,0,0,0,0,9,46,26,2,0,0,0,0,15,52,48,35,0,0,0,0,0,0,0,0,47,28,48,48,0,0,0,0,30,14,55,54,0,0,0,0,13,2,17,30,0,0,0,0,15,59,31,44],[1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,47,0,3,58,0,0,0,0,0,47,3,0,0,0,0,0,0,36,14,0,0,0,0,0,25,36,0,14],[2,26,46,9,0,0,0,0,35,48,52,15,0,0,0,0,48,35,59,35,0,0,0,0,26,2,26,13,0,0,0,0,0,0,0,0,54,14,43,43,0,0,0,0,39,7,56,48,0,0,0,0,25,17,15,39,0,0,0,0,42,44,22,46] >;
57 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 | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20F | 20G | ··· | 20L | 30A | 30B | 60A | ··· | 60F |
| 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 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 10 | 10 | 10 | 30 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | 4 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | 2- (1+4) | S3×D5 | Q8○D12 | C2×S3×D5 | Q8.10D10 | D20.29D6 |
| kernel | D20.29D6 | D5×Dic6 | D20⋊5S3 | D20⋊S3 | D6.D10 | C3×Q8⋊2D5 | C5×S3×Q8 | Q8×D15 | Q8⋊2D5 | S3×Q8 | C4×D5 | D20 | C5×Q8 | Dic6 | C4×S3 | C3×Q8 | C15 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 1 | 2 | 2 | 6 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{20}._{29}D_6 % in TeX
G:=Group("D20.29D6"); // GroupNames label
G:=SmallGroup(480,1104);
// by ID
G=gap.SmallGroup(480,1104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=a^10*c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀