metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.24D6, C60.17C23, Dic6.8D10, Dic30.5C22, D4⋊D5⋊7S3, D4.3(S3×D5), D4⋊2S3⋊2D5, C15⋊15(C4○D8), C5⋊5(D8⋊3S3), D4.D15⋊7C2, C5⋊Dic12⋊3C2, (C3×D4).5D10, (C5×D4).20D6, C5⋊2C8.13D6, D20⋊5S3⋊1C2, (S3×C10).10D4, (C4×S3).21D10, C10.143(S3×D4), C30.179(C2×D4), C30.D4⋊4C2, D6.1(C5⋊D4), C3⋊2(D4.8D10), (S3×C20).5C22, C20.17(C22×S3), C15⋊3C8.5C22, (C5×Dic3).36D4, (C3×D20).6C22, C12.17(C22×D5), (D4×C15).11C22, (C5×Dic6).5C22, Dic3.20(C5⋊D4), C4.17(C2×S3×D5), (S3×C5⋊2C8)⋊3C2, (C3×D4⋊D5)⋊5C2, C2.24(S3×C5⋊D4), C6.46(C2×C5⋊D4), (C5×D4⋊2S3)⋊2C2, (C3×C5⋊2C8).3C22, SmallGroup(480,569)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.24D6
G = < a,b,c,d | a20=b2=c6=1, d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=c-1 >
Subgroups: 588 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], D5, C10, C10 [×2], Dic3, Dic3 [×2], C12, D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, C2×C10 [×2], C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×D4, C5×S3, C3×D5, C30, C30, C4○D8, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, S3×C8, Dic12, D4.S3 [×2], C3×D8, D4⋊2S3, D4⋊2S3, C5×Dic3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, C2×C30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, D8⋊3S3, C3×C5⋊2C8, C15⋊3C8, D5×Dic3, C15⋊D4, C3×D20, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, Dic30, D4×C15, D4.8D10, S3×C5⋊2C8, C30.D4, C5⋊Dic12, C3×D4⋊D5, D4.D15, D20⋊5S3, C5×D4⋊2S3, D20.24D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊3S3, C2×S3×D5, D4.8D10, S3×C5⋊D4, D20.24D6
►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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 220)(22 219)(23 218)(24 217)(25 216)(26 215)(27 214)(28 213)(29 212)(30 211)(31 210)(32 209)(33 208)(34 207)(35 206)(36 205)(37 204)(38 203)(39 202)(40 201)(61 177)(62 176)(63 175)(64 174)(65 173)(66 172)(67 171)(68 170)(69 169)(70 168)(71 167)(72 166)(73 165)(74 164)(75 163)(76 162)(77 161)(78 180)(79 179)(80 178)(81 239)(82 238)(83 237)(84 236)(85 235)(86 234)(87 233)(88 232)(89 231)(90 230)(91 229)(92 228)(93 227)(94 226)(95 225)(96 224)(97 223)(98 222)(99 221)(100 240)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 195)(122 194)(123 193)(124 192)(125 191)(126 190)(127 189)(128 188)(129 187)(130 186)(131 185)(132 184)(133 183)(134 182)(135 181)(136 200)(137 199)(138 198)(139 197)(140 196)
(1 108 82 132 168 25)(2 119 83 123 169 36)(3 110 84 134 170 27)(4 101 85 125 171 38)(5 112 86 136 172 29)(6 103 87 127 173 40)(7 114 88 138 174 31)(8 105 89 129 175 22)(9 116 90 140 176 33)(10 107 91 131 177 24)(11 118 92 122 178 35)(12 109 93 133 179 26)(13 120 94 124 180 37)(14 111 95 135 161 28)(15 102 96 126 162 39)(16 113 97 137 163 30)(17 104 98 128 164 21)(18 115 99 139 165 32)(19 106 100 130 166 23)(20 117 81 121 167 34)(41 144 225 186 77 218)(42 155 226 197 78 209)(43 146 227 188 79 220)(44 157 228 199 80 211)(45 148 229 190 61 202)(46 159 230 181 62 213)(47 150 231 192 63 204)(48 141 232 183 64 215)(49 152 233 194 65 206)(50 143 234 185 66 217)(51 154 235 196 67 208)(52 145 236 187 68 219)(53 156 237 198 69 210)(54 147 238 189 70 201)(55 158 239 200 71 212)(56 149 240 191 72 203)(57 160 221 182 73 214)(58 151 222 193 74 205)(59 142 223 184 75 216)(60 153 224 195 76 207)
(1 30 11 40)(2 21 12 31)(3 32 13 22)(4 23 14 33)(5 34 15 24)(6 25 16 35)(7 36 17 26)(8 27 18 37)(9 38 19 28)(10 29 20 39)(41 213 51 203)(42 204 52 214)(43 215 53 205)(44 206 54 216)(45 217 55 207)(46 208 56 218)(47 219 57 209)(48 210 58 220)(49 201 59 211)(50 212 60 202)(61 143 71 153)(62 154 72 144)(63 145 73 155)(64 156 74 146)(65 147 75 157)(66 158 76 148)(67 149 77 159)(68 160 78 150)(69 151 79 141)(70 142 80 152)(81 126 91 136)(82 137 92 127)(83 128 93 138)(84 139 94 129)(85 130 95 140)(86 121 96 131)(87 132 97 122)(88 123 98 133)(89 134 99 124)(90 125 100 135)(101 166 111 176)(102 177 112 167)(103 168 113 178)(104 179 114 169)(105 170 115 180)(106 161 116 171)(107 172 117 162)(108 163 118 173)(109 174 119 164)(110 165 120 175)(181 235 191 225)(182 226 192 236)(183 237 193 227)(184 228 194 238)(185 239 195 229)(186 230 196 240)(187 221 197 231)(188 232 198 222)(189 223 199 233)(190 234 200 224)
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,220)(22,219)(23,218)(24,217)(25,216)(26,215)(27,214)(28,213)(29,212)(30,211)(31,210)(32,209)(33,208)(34,207)(35,206)(36,205)(37,204)(38,203)(39,202)(40,201)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,180)(79,179)(80,178)(81,239)(82,238)(83,237)(84,236)(85,235)(86,234)(87,233)(88,232)(89,231)(90,230)(91,229)(92,228)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,240)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,195)(122,194)(123,193)(124,192)(125,191)(126,190)(127,189)(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,200)(137,199)(138,198)(139,197)(140,196), (1,108,82,132,168,25)(2,119,83,123,169,36)(3,110,84,134,170,27)(4,101,85,125,171,38)(5,112,86,136,172,29)(6,103,87,127,173,40)(7,114,88,138,174,31)(8,105,89,129,175,22)(9,116,90,140,176,33)(10,107,91,131,177,24)(11,118,92,122,178,35)(12,109,93,133,179,26)(13,120,94,124,180,37)(14,111,95,135,161,28)(15,102,96,126,162,39)(16,113,97,137,163,30)(17,104,98,128,164,21)(18,115,99,139,165,32)(19,106,100,130,166,23)(20,117,81,121,167,34)(41,144,225,186,77,218)(42,155,226,197,78,209)(43,146,227,188,79,220)(44,157,228,199,80,211)(45,148,229,190,61,202)(46,159,230,181,62,213)(47,150,231,192,63,204)(48,141,232,183,64,215)(49,152,233,194,65,206)(50,143,234,185,66,217)(51,154,235,196,67,208)(52,145,236,187,68,219)(53,156,237,198,69,210)(54,147,238,189,70,201)(55,158,239,200,71,212)(56,149,240,191,72,203)(57,160,221,182,73,214)(58,151,222,193,74,205)(59,142,223,184,75,216)(60,153,224,195,76,207), (1,30,11,40)(2,21,12,31)(3,32,13,22)(4,23,14,33)(5,34,15,24)(6,25,16,35)(7,36,17,26)(8,27,18,37)(9,38,19,28)(10,29,20,39)(41,213,51,203)(42,204,52,214)(43,215,53,205)(44,206,54,216)(45,217,55,207)(46,208,56,218)(47,219,57,209)(48,210,58,220)(49,201,59,211)(50,212,60,202)(61,143,71,153)(62,154,72,144)(63,145,73,155)(64,156,74,146)(65,147,75,157)(66,158,76,148)(67,149,77,159)(68,160,78,150)(69,151,79,141)(70,142,80,152)(81,126,91,136)(82,137,92,127)(83,128,93,138)(84,139,94,129)(85,130,95,140)(86,121,96,131)(87,132,97,122)(88,123,98,133)(89,134,99,124)(90,125,100,135)(101,166,111,176)(102,177,112,167)(103,168,113,178)(104,179,114,169)(105,170,115,180)(106,161,116,171)(107,172,117,162)(108,163,118,173)(109,174,119,164)(110,165,120,175)(181,235,191,225)(182,226,192,236)(183,237,193,227)(184,228,194,238)(185,239,195,229)(186,230,196,240)(187,221,197,231)(188,232,198,222)(189,223,199,233)(190,234,200,224)>;
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,220)(22,219)(23,218)(24,217)(25,216)(26,215)(27,214)(28,213)(29,212)(30,211)(31,210)(32,209)(33,208)(34,207)(35,206)(36,205)(37,204)(38,203)(39,202)(40,201)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,180)(79,179)(80,178)(81,239)(82,238)(83,237)(84,236)(85,235)(86,234)(87,233)(88,232)(89,231)(90,230)(91,229)(92,228)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,240)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,195)(122,194)(123,193)(124,192)(125,191)(126,190)(127,189)(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,200)(137,199)(138,198)(139,197)(140,196), (1,108,82,132,168,25)(2,119,83,123,169,36)(3,110,84,134,170,27)(4,101,85,125,171,38)(5,112,86,136,172,29)(6,103,87,127,173,40)(7,114,88,138,174,31)(8,105,89,129,175,22)(9,116,90,140,176,33)(10,107,91,131,177,24)(11,118,92,122,178,35)(12,109,93,133,179,26)(13,120,94,124,180,37)(14,111,95,135,161,28)(15,102,96,126,162,39)(16,113,97,137,163,30)(17,104,98,128,164,21)(18,115,99,139,165,32)(19,106,100,130,166,23)(20,117,81,121,167,34)(41,144,225,186,77,218)(42,155,226,197,78,209)(43,146,227,188,79,220)(44,157,228,199,80,211)(45,148,229,190,61,202)(46,159,230,181,62,213)(47,150,231,192,63,204)(48,141,232,183,64,215)(49,152,233,194,65,206)(50,143,234,185,66,217)(51,154,235,196,67,208)(52,145,236,187,68,219)(53,156,237,198,69,210)(54,147,238,189,70,201)(55,158,239,200,71,212)(56,149,240,191,72,203)(57,160,221,182,73,214)(58,151,222,193,74,205)(59,142,223,184,75,216)(60,153,224,195,76,207), (1,30,11,40)(2,21,12,31)(3,32,13,22)(4,23,14,33)(5,34,15,24)(6,25,16,35)(7,36,17,26)(8,27,18,37)(9,38,19,28)(10,29,20,39)(41,213,51,203)(42,204,52,214)(43,215,53,205)(44,206,54,216)(45,217,55,207)(46,208,56,218)(47,219,57,209)(48,210,58,220)(49,201,59,211)(50,212,60,202)(61,143,71,153)(62,154,72,144)(63,145,73,155)(64,156,74,146)(65,147,75,157)(66,158,76,148)(67,149,77,159)(68,160,78,150)(69,151,79,141)(70,142,80,152)(81,126,91,136)(82,137,92,127)(83,128,93,138)(84,139,94,129)(85,130,95,140)(86,121,96,131)(87,132,97,122)(88,123,98,133)(89,134,99,124)(90,125,100,135)(101,166,111,176)(102,177,112,167)(103,168,113,178)(104,179,114,169)(105,170,115,180)(106,161,116,171)(107,172,117,162)(108,163,118,173)(109,174,119,164)(110,165,120,175)(181,235,191,225)(182,226,192,236)(183,237,193,227)(184,228,194,238)(185,239,195,229)(186,230,196,240)(187,221,197,231)(188,232,198,222)(189,223,199,233)(190,234,200,224) );
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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,60),(16,59),(17,58),(18,57),(19,56),(20,55),(21,220),(22,219),(23,218),(24,217),(25,216),(26,215),(27,214),(28,213),(29,212),(30,211),(31,210),(32,209),(33,208),(34,207),(35,206),(36,205),(37,204),(38,203),(39,202),(40,201),(61,177),(62,176),(63,175),(64,174),(65,173),(66,172),(67,171),(68,170),(69,169),(70,168),(71,167),(72,166),(73,165),(74,164),(75,163),(76,162),(77,161),(78,180),(79,179),(80,178),(81,239),(82,238),(83,237),(84,236),(85,235),(86,234),(87,233),(88,232),(89,231),(90,230),(91,229),(92,228),(93,227),(94,226),(95,225),(96,224),(97,223),(98,222),(99,221),(100,240),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,160),(111,159),(112,158),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,195),(122,194),(123,193),(124,192),(125,191),(126,190),(127,189),(128,188),(129,187),(130,186),(131,185),(132,184),(133,183),(134,182),(135,181),(136,200),(137,199),(138,198),(139,197),(140,196)], [(1,108,82,132,168,25),(2,119,83,123,169,36),(3,110,84,134,170,27),(4,101,85,125,171,38),(5,112,86,136,172,29),(6,103,87,127,173,40),(7,114,88,138,174,31),(8,105,89,129,175,22),(9,116,90,140,176,33),(10,107,91,131,177,24),(11,118,92,122,178,35),(12,109,93,133,179,26),(13,120,94,124,180,37),(14,111,95,135,161,28),(15,102,96,126,162,39),(16,113,97,137,163,30),(17,104,98,128,164,21),(18,115,99,139,165,32),(19,106,100,130,166,23),(20,117,81,121,167,34),(41,144,225,186,77,218),(42,155,226,197,78,209),(43,146,227,188,79,220),(44,157,228,199,80,211),(45,148,229,190,61,202),(46,159,230,181,62,213),(47,150,231,192,63,204),(48,141,232,183,64,215),(49,152,233,194,65,206),(50,143,234,185,66,217),(51,154,235,196,67,208),(52,145,236,187,68,219),(53,156,237,198,69,210),(54,147,238,189,70,201),(55,158,239,200,71,212),(56,149,240,191,72,203),(57,160,221,182,73,214),(58,151,222,193,74,205),(59,142,223,184,75,216),(60,153,224,195,76,207)], [(1,30,11,40),(2,21,12,31),(3,32,13,22),(4,23,14,33),(5,34,15,24),(6,25,16,35),(7,36,17,26),(8,27,18,37),(9,38,19,28),(10,29,20,39),(41,213,51,203),(42,204,52,214),(43,215,53,205),(44,206,54,216),(45,217,55,207),(46,208,56,218),(47,219,57,209),(48,210,58,220),(49,201,59,211),(50,212,60,202),(61,143,71,153),(62,154,72,144),(63,145,73,155),(64,156,74,146),(65,147,75,157),(66,158,76,148),(67,149,77,159),(68,160,78,150),(69,151,79,141),(70,142,80,152),(81,126,91,136),(82,137,92,127),(83,128,93,138),(84,139,94,129),(85,130,95,140),(86,121,96,131),(87,132,97,122),(88,123,98,133),(89,134,99,124),(90,125,100,135),(101,166,111,176),(102,177,112,167),(103,168,113,178),(104,179,114,169),(105,170,115,180),(106,161,116,171),(107,172,117,162),(108,163,118,173),(109,174,119,164),(110,165,120,175),(181,235,191,225),(182,226,192,236),(183,237,193,227),(184,228,194,238),(185,239,195,229),(186,230,196,240),(187,221,197,231),(188,232,198,222),(189,223,199,233),(190,234,200,224)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 6 | 20 | 2 | 2 | 3 | 3 | 12 | 60 | 2 | 2 | 2 | 8 | 40 | 10 | 10 | 30 | 30 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
51 irreducible representations
| dim | 1 | 1 | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | D8⋊3S3 | C2×S3×D5 | D4.8D10 | S3×C5⋊D4 | D20.24D6 |
| kernel | D20.24D6 | S3×C5⋊2C8 | C30.D4 | C5⋊Dic12 | C3×D4⋊D5 | D4.D15 | D20⋊5S3 | C5×D4⋊2S3 | D4⋊D5 | C5×Dic3 | S3×C10 | D4⋊2S3 | C5⋊2C8 | D20 | C5×D4 | Dic6 | C4×S3 | C3×D4 | C15 | Dic3 | D6 | C10 | D4 | 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 | 4 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.24D6 ►in GL6(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 211 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 211 | 177 | 0 | 0 |
| 0 | 0 | 63 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,0,177,0,0,0,0,0,0,189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,211,0,0,0,0,0,0,0,211,63,0,0,0,0,177,30,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[0,177,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,1,1] >;
D20.24D6 in GAP, Magma, Sage, TeX
D_{20}._{24}D_6 % in TeX
G:=Group("D20.24D6"); // GroupNames label
G:=SmallGroup(480,569);
// by ID
G=gap.SmallGroup(480,569);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations