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G = D20.D6order 480 = 25·3·5

15th non-split extension by D20 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.15D6, C60.40C23, Dic6.28D10, D60.14C22, C3⋊D407C2, C3⋊C8.20D10, C3⋊Q167D5, C1520(C4○D8), Q82D56S3, (C4×D5).50D6, (C6×D5).15D4, C6.153(D4×D5), Q8.19(S3×D5), (C5×Q8).25D6, C12.28D103C2, C35(Q8.D10), Q82D156C2, C30.202(C2×D4), C30.D47C2, C55(Q8.13D6), (C3×Q8).23D10, C20.40(C22×S3), (C3×Dic5).73D4, C12.40(C22×D5), D10.12(C3⋊D4), C153C8.14C22, (D5×C12).16C22, (C3×D20).14C22, (Q8×C15).10C22, Dic5.43(C3⋊D4), (C5×Dic6).14C22, (D5×C3⋊C8)⋊7C2, C4.40(C2×S3×D5), (C5×C3⋊Q16)⋊6C2, C2.35(D5×C3⋊D4), (C3×Q82D5)⋊3C2, C10.56(C2×C3⋊D4), (C5×C3⋊C8).14C22, SmallGroup(480,592)

Series: Derived Chief Lower central Upper central

C1C60 — D20.D6
C1C5C15C30C60D5×C12C12.28D10 — D20.D6
C15C30C60 — D20.D6
C1C2C4Q8

Generators and relations for D20.D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a13b, dcd=a10c-1 >

Subgroups: 700 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 [×4], Q8, Q8, D5 [×3], C10, Dic3, C12, C12 [×2], D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, D10 [×2], C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×Q8, C3×D5 [×2], D15, C30, C4○D8, C52C8, C40, C4×D5, C4×D5 [×2], D20, D20 [×3], C5×Q8, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C60, C6×D5, C6×D5, D30, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q82D5, Q82D5, Q8.13D6, C5×C3⋊C8, C153C8, D30.C2, C3⋊D20, D5×C12, D5×C12, C3×D20, C3×D20, C5×Dic6, D60, Q8×C15, Q8.D10, D5×C3⋊C8, C3⋊D40, C30.D4, C5×C3⋊Q16, Q82D15, C12.28D10, C3×Q82D5, D20.D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, Q8.D10, D5×C3⋊D4, D20.D6

Smallest permutation representation of D20.D6
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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 140)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 111)(38 110)(39 109)(40 108)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)(53 100)(54 99)(55 98)(56 97)(57 96)(58 95)(59 94)(60 93)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(161 238)(162 237)(163 236)(164 235)(165 234)(166 233)(167 232)(168 231)(169 230)(170 229)(171 228)(172 227)(173 226)(174 225)(175 224)(176 223)(177 222)(178 221)(179 240)(180 239)(181 205)(182 204)(183 203)(184 202)(185 201)(186 220)(187 219)(188 218)(189 217)(190 216)(191 215)(192 214)(193 213)(194 212)(195 211)(196 210)(197 209)(198 208)(199 207)(200 206)
(1 170 154)(2 179 155 10 171 143)(3 168 156 19 172 152)(4 177 157 8 173 141)(5 166 158 17 174 150)(6 175 159)(7 164 160 15 176 148)(9 162 142 13 178 146)(11 180 144)(12 169 145 20 161 153)(14 167 147 18 163 151)(16 165 149)(21 213 42 39 215 60)(22 202 43 28 216 49)(23 211 44 37 217 58)(24 220 45 26 218 47)(25 209 46 35 219 56)(27 207 48 33 201 54)(29 205 50 31 203 52)(30 214 51 40 204 41)(32 212 53 38 206 59)(34 210 55 36 208 57)(61 124 236 79 126 234)(62 133 237 68 127 223)(63 122 238 77 128 232)(64 131 239 66 129 221)(65 140 240 75 130 230)(67 138 222 73 132 228)(69 136 224 71 134 226)(70 125 225 80 135 235)(72 123 227 78 137 233)(74 121 229 76 139 231)(81 101 181 85 117 185)(82 110 182 94 118 194)(83 119 183)(84 108 184 92 120 192)(86 106 186 90 102 190)(87 115 187 99 103 199)(88 104 188)(89 113 189 97 105 197)(91 111 191 95 107 195)(93 109 193)(96 116 196 100 112 200)(98 114 198)
(1 214)(2 213)(3 212)(4 211)(5 210)(6 209)(7 208)(8 207)(9 206)(10 205)(11 204)(12 203)(13 202)(14 201)(15 220)(16 219)(17 218)(18 217)(19 216)(20 215)(21 169)(22 168)(23 167)(24 166)(25 165)(26 164)(27 163)(28 162)(29 161)(30 180)(31 179)(32 178)(33 177)(34 176)(35 175)(36 174)(37 173)(38 172)(39 171)(40 170)(41 154)(42 153)(43 152)(44 151)(45 150)(46 149)(47 148)(48 147)(49 146)(50 145)(51 144)(52 143)(53 142)(54 141)(55 160)(56 159)(57 158)(58 157)(59 156)(60 155)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(101 223)(102 222)(103 221)(104 240)(105 239)(106 238)(107 237)(108 236)(109 235)(110 234)(111 233)(112 232)(113 231)(114 230)(115 229)(116 228)(117 227)(118 226)(119 225)(120 224)(121 197)(122 196)(123 195)(124 194)(125 193)(126 192)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(136 182)(137 181)(138 200)(139 199)(140 198)

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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(161,238)(162,237)(163,236)(164,235)(165,234)(166,233)(167,232)(168,231)(169,230)(170,229)(171,228)(172,227)(173,226)(174,225)(175,224)(176,223)(177,222)(178,221)(179,240)(180,239)(181,205)(182,204)(183,203)(184,202)(185,201)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,210)(197,209)(198,208)(199,207)(200,206), (1,170,154)(2,179,155,10,171,143)(3,168,156,19,172,152)(4,177,157,8,173,141)(5,166,158,17,174,150)(6,175,159)(7,164,160,15,176,148)(9,162,142,13,178,146)(11,180,144)(12,169,145,20,161,153)(14,167,147,18,163,151)(16,165,149)(21,213,42,39,215,60)(22,202,43,28,216,49)(23,211,44,37,217,58)(24,220,45,26,218,47)(25,209,46,35,219,56)(27,207,48,33,201,54)(29,205,50,31,203,52)(30,214,51,40,204,41)(32,212,53,38,206,59)(34,210,55,36,208,57)(61,124,236,79,126,234)(62,133,237,68,127,223)(63,122,238,77,128,232)(64,131,239,66,129,221)(65,140,240,75,130,230)(67,138,222,73,132,228)(69,136,224,71,134,226)(70,125,225,80,135,235)(72,123,227,78,137,233)(74,121,229,76,139,231)(81,101,181,85,117,185)(82,110,182,94,118,194)(83,119,183)(84,108,184,92,120,192)(86,106,186,90,102,190)(87,115,187,99,103,199)(88,104,188)(89,113,189,97,105,197)(91,111,191,95,107,195)(93,109,193)(96,116,196,100,112,200)(98,114,198), (1,214)(2,213)(3,212)(4,211)(5,210)(6,209)(7,208)(8,207)(9,206)(10,205)(11,204)(12,203)(13,202)(14,201)(15,220)(16,219)(17,218)(18,217)(19,216)(20,215)(21,169)(22,168)(23,167)(24,166)(25,165)(26,164)(27,163)(28,162)(29,161)(30,180)(31,179)(32,178)(33,177)(34,176)(35,175)(36,174)(37,173)(38,172)(39,171)(40,170)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,160)(56,159)(57,158)(58,157)(59,156)(60,155)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(101,223)(102,222)(103,221)(104,240)(105,239)(106,238)(107,237)(108,236)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,228)(117,227)(118,226)(119,225)(120,224)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,200)(139,199)(140,198)>;

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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(161,238)(162,237)(163,236)(164,235)(165,234)(166,233)(167,232)(168,231)(169,230)(170,229)(171,228)(172,227)(173,226)(174,225)(175,224)(176,223)(177,222)(178,221)(179,240)(180,239)(181,205)(182,204)(183,203)(184,202)(185,201)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,210)(197,209)(198,208)(199,207)(200,206), (1,170,154)(2,179,155,10,171,143)(3,168,156,19,172,152)(4,177,157,8,173,141)(5,166,158,17,174,150)(6,175,159)(7,164,160,15,176,148)(9,162,142,13,178,146)(11,180,144)(12,169,145,20,161,153)(14,167,147,18,163,151)(16,165,149)(21,213,42,39,215,60)(22,202,43,28,216,49)(23,211,44,37,217,58)(24,220,45,26,218,47)(25,209,46,35,219,56)(27,207,48,33,201,54)(29,205,50,31,203,52)(30,214,51,40,204,41)(32,212,53,38,206,59)(34,210,55,36,208,57)(61,124,236,79,126,234)(62,133,237,68,127,223)(63,122,238,77,128,232)(64,131,239,66,129,221)(65,140,240,75,130,230)(67,138,222,73,132,228)(69,136,224,71,134,226)(70,125,225,80,135,235)(72,123,227,78,137,233)(74,121,229,76,139,231)(81,101,181,85,117,185)(82,110,182,94,118,194)(83,119,183)(84,108,184,92,120,192)(86,106,186,90,102,190)(87,115,187,99,103,199)(88,104,188)(89,113,189,97,105,197)(91,111,191,95,107,195)(93,109,193)(96,116,196,100,112,200)(98,114,198), (1,214)(2,213)(3,212)(4,211)(5,210)(6,209)(7,208)(8,207)(9,206)(10,205)(11,204)(12,203)(13,202)(14,201)(15,220)(16,219)(17,218)(18,217)(19,216)(20,215)(21,169)(22,168)(23,167)(24,166)(25,165)(26,164)(27,163)(28,162)(29,161)(30,180)(31,179)(32,178)(33,177)(34,176)(35,175)(36,174)(37,173)(38,172)(39,171)(40,170)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,160)(56,159)(57,158)(58,157)(59,156)(60,155)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(101,223)(102,222)(103,221)(104,240)(105,239)(106,238)(107,237)(108,236)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,228)(117,227)(118,226)(119,225)(120,224)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,200)(139,199)(140,198) );

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)], [(1,139),(2,138),(3,137),(4,136),(5,135),(6,134),(7,133),(8,132),(9,131),(10,130),(11,129),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,121),(20,140),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,111),(38,110),(39,109),(40,108),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,84),(50,83),(51,82),(52,81),(53,100),(54,99),(55,98),(56,97),(57,96),(58,95),(59,94),(60,93),(61,147),(62,146),(63,145),(64,144),(65,143),(66,142),(67,141),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,154),(75,153),(76,152),(77,151),(78,150),(79,149),(80,148),(161,238),(162,237),(163,236),(164,235),(165,234),(166,233),(167,232),(168,231),(169,230),(170,229),(171,228),(172,227),(173,226),(174,225),(175,224),(176,223),(177,222),(178,221),(179,240),(180,239),(181,205),(182,204),(183,203),(184,202),(185,201),(186,220),(187,219),(188,218),(189,217),(190,216),(191,215),(192,214),(193,213),(194,212),(195,211),(196,210),(197,209),(198,208),(199,207),(200,206)], [(1,170,154),(2,179,155,10,171,143),(3,168,156,19,172,152),(4,177,157,8,173,141),(5,166,158,17,174,150),(6,175,159),(7,164,160,15,176,148),(9,162,142,13,178,146),(11,180,144),(12,169,145,20,161,153),(14,167,147,18,163,151),(16,165,149),(21,213,42,39,215,60),(22,202,43,28,216,49),(23,211,44,37,217,58),(24,220,45,26,218,47),(25,209,46,35,219,56),(27,207,48,33,201,54),(29,205,50,31,203,52),(30,214,51,40,204,41),(32,212,53,38,206,59),(34,210,55,36,208,57),(61,124,236,79,126,234),(62,133,237,68,127,223),(63,122,238,77,128,232),(64,131,239,66,129,221),(65,140,240,75,130,230),(67,138,222,73,132,228),(69,136,224,71,134,226),(70,125,225,80,135,235),(72,123,227,78,137,233),(74,121,229,76,139,231),(81,101,181,85,117,185),(82,110,182,94,118,194),(83,119,183),(84,108,184,92,120,192),(86,106,186,90,102,190),(87,115,187,99,103,199),(88,104,188),(89,113,189,97,105,197),(91,111,191,95,107,195),(93,109,193),(96,116,196,100,112,200),(98,114,198)], [(1,214),(2,213),(3,212),(4,211),(5,210),(6,209),(7,208),(8,207),(9,206),(10,205),(11,204),(12,203),(13,202),(14,201),(15,220),(16,219),(17,218),(18,217),(19,216),(20,215),(21,169),(22,168),(23,167),(24,166),(25,165),(26,164),(27,163),(28,162),(29,161),(30,180),(31,179),(32,178),(33,177),(34,176),(35,175),(36,174),(37,173),(38,172),(39,171),(40,170),(41,154),(42,153),(43,152),(44,151),(45,150),(46,149),(47,148),(48,147),(49,146),(50,145),(51,144),(52,143),(53,142),(54,141),(55,160),(56,159),(57,158),(58,157),(59,156),(60,155),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(101,223),(102,222),(103,221),(104,240),(105,239),(106,238),(107,237),(108,236),(109,235),(110,234),(111,233),(112,232),(113,231),(114,230),(115,229),(116,228),(117,227),(118,226),(119,225),(120,224),(121,197),(122,196),(123,195),(124,194),(125,193),(126,192),(127,191),(128,190),(129,189),(130,188),(131,187),(132,186),(133,185),(134,184),(135,183),(136,182),(137,181),(138,200),(139,199),(140,198)])

48 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C6D8A8B8C8D10A10B12A12B12C12D12E15A15B20A20B20C20D20E20F30A30B40A40B40C40D60A···60F
order12222344444556666888810101212121212151520202020202030304040404060···60
size111020602245512222202020663030224441010444488242444121212128···8

48 irreducible representations

dim1111111122222222222224444448
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4C4○D8S3×D5D4×D5Q8.13D6C2×S3×D5Q8.D10D5×C3⋊D4D20.D6
kernelD20.D6D5×C3⋊C8C3⋊D40C30.D4C5×C3⋊Q16Q82D15C12.28D10C3×Q82D5Q82D5C3×Dic5C6×D5C3⋊Q16C4×D5D20C5×Q8C3⋊C8Dic6C3×Q8Dic5D10C15Q8C6C5C4C3C2C1
# reps1111111111121112222242222442

Matrix representation of D20.D6 in GL6(𝔽241)

6400000
1431770000
000100
0024018900
00002400
00000240
,
92380000
1072320000
000100
001000
0000210112
000019831
,
100000
62400000
001000
0018924000
0000149
0000177239
,
212900000
71290000
00240000
0052100
000012176
000023120

G:=sub<GL(6,GF(241))| [64,143,0,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,1,189,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[9,107,0,0,0,0,238,232,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,210,198,0,0,0,0,112,31],[1,6,0,0,0,0,0,240,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,177,0,0,0,0,49,239],[212,71,0,0,0,0,90,29,0,0,0,0,0,0,240,52,0,0,0,0,0,1,0,0,0,0,0,0,121,23,0,0,0,0,76,120] >;

D20.D6 in GAP, Magma, Sage, TeX

D_{20}.D_6
% in TeX

G:=Group("D20.D6");
// GroupNames label

G:=SmallGroup(480,592);
// by ID

G=gap.SmallGroup(480,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,422,135,100,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^13*b,d*c*d=a^10*c^-1>;
// generators/relations

׿
×
𝔽