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G = D6.D20order 480 = 25·3·5

5th non-split extension by D6 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.8D20, D6⋊C417D5, C4⋊Dic55S3, (S3×C10).6D4, C10.17(S3×D4), (C2×C20).21D6, C6.17(C2×D20), C2.19(S3×D20), C30.48(C2×D4), C53(D6.D4), D303C415C2, D304C415C2, C30.68(C4○D4), (C2×C12).226D10, C6.Dic1021C2, C10.70(C4○D12), C6.45(D42D5), (C2×C30).117C23, (C2×C60).257C22, (C2×Dic5).111D6, (C2×Dic3).36D10, (C22×S3).41D10, C31(C22.D20), C2.18(D12⋊D5), C10.35(Q83S3), C1510(C22.D4), (C6×Dic5).70C22, C2.16(Dic3.D10), (C10×Dic3).72C22, (C2×Dic15).95C22, (C22×D15).39C22, (C2×S3×Dic5)⋊6C2, (C5×D6⋊C4)⋊17C2, (C2×C4).51(S3×D5), (C3×C4⋊Dic5)⋊16C2, (C2×C5⋊D12).7C2, C22.181(C2×S3×D5), (S3×C2×C10).22C22, (C2×C6).129(C22×D5), (C2×C10).129(C22×S3), SmallGroup(480,503)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D6.D20
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — D6.D20
C15C2×C30 — D6.D20
C1C22C2×C4

Generators and relations for D6.D20
 G = < a,b,c,d | a6=b2=c20=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 908 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×3], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5, C10 [×3], C10 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×4], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3 [×2], D15, C30 [×3], C22.D4, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5 [×2], Dic15, C60, S3×C10 [×2], S3×C10 [×2], D30 [×3], C2×C30, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, D6.D4, S3×Dic5 [×2], C5⋊D12 [×2], C6×Dic5 [×2], C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C22.D20, D304C4, C6.Dic10, C3×C4⋊Dic5, C5×D6⋊C4, D303C4, C2×S3×Dic5, C2×C5⋊D12, D6.D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C22.D4, D20 [×2], C22×D5, C4○D12, S3×D4, Q83S3, S3×D5, C2×D20, D42D5 [×2], D6.D4, C2×S3×D5, C22.D20, D12⋊D5, S3×D20, Dic3.D10, D6.D20

Smallest permutation representation of D6.D20
On 240 points
Generators in S240
(1 122 217 62 155 176)(2 123 218 63 156 177)(3 124 219 64 157 178)(4 125 220 65 158 179)(5 126 201 66 159 180)(6 127 202 67 160 161)(7 128 203 68 141 162)(8 129 204 69 142 163)(9 130 205 70 143 164)(10 131 206 71 144 165)(11 132 207 72 145 166)(12 133 208 73 146 167)(13 134 209 74 147 168)(14 135 210 75 148 169)(15 136 211 76 149 170)(16 137 212 77 150 171)(17 138 213 78 151 172)(18 139 214 79 152 173)(19 140 215 80 153 174)(20 121 216 61 154 175)(21 108 228 188 99 43)(22 109 229 189 100 44)(23 110 230 190 81 45)(24 111 231 191 82 46)(25 112 232 192 83 47)(26 113 233 193 84 48)(27 114 234 194 85 49)(28 115 235 195 86 50)(29 116 236 196 87 51)(30 117 237 197 88 52)(31 118 238 198 89 53)(32 119 239 199 90 54)(33 120 240 200 91 55)(34 101 221 181 92 56)(35 102 222 182 93 57)(36 103 223 183 94 58)(37 104 224 184 95 59)(38 105 225 185 96 60)(39 106 226 186 97 41)(40 107 227 187 98 42)
(1 176)(2 218)(3 178)(4 220)(5 180)(6 202)(7 162)(8 204)(9 164)(10 206)(11 166)(12 208)(13 168)(14 210)(15 170)(16 212)(17 172)(18 214)(19 174)(20 216)(21 108)(22 100)(23 110)(24 82)(25 112)(26 84)(27 114)(28 86)(29 116)(30 88)(31 118)(32 90)(33 120)(34 92)(35 102)(36 94)(37 104)(38 96)(39 106)(40 98)(41 226)(43 228)(45 230)(47 232)(49 234)(51 236)(53 238)(55 240)(57 222)(59 224)(61 175)(62 217)(63 177)(64 219)(65 179)(66 201)(67 161)(68 203)(69 163)(70 205)(71 165)(72 207)(73 167)(74 209)(75 169)(76 211)(77 171)(78 213)(79 173)(80 215)(81 190)(83 192)(85 194)(87 196)(89 198)(91 200)(93 182)(95 184)(97 186)(99 188)(101 181)(103 183)(105 185)(107 187)(109 189)(111 191)(113 193)(115 195)(117 197)(119 199)(122 155)(124 157)(126 159)(128 141)(130 143)(132 145)(134 147)(136 149)(138 151)(140 153)
(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)
(1 115 62 86)(2 114 63 85)(3 113 64 84)(4 112 65 83)(5 111 66 82)(6 110 67 81)(7 109 68 100)(8 108 69 99)(9 107 70 98)(10 106 71 97)(11 105 72 96)(12 104 73 95)(13 103 74 94)(14 102 75 93)(15 101 76 92)(16 120 77 91)(17 119 78 90)(18 118 79 89)(19 117 80 88)(20 116 61 87)(21 204 188 163)(22 203 189 162)(23 202 190 161)(24 201 191 180)(25 220 192 179)(26 219 193 178)(27 218 194 177)(28 217 195 176)(29 216 196 175)(30 215 197 174)(31 214 198 173)(32 213 199 172)(33 212 200 171)(34 211 181 170)(35 210 182 169)(36 209 183 168)(37 208 184 167)(38 207 185 166)(39 206 186 165)(40 205 187 164)(41 131 226 144)(42 130 227 143)(43 129 228 142)(44 128 229 141)(45 127 230 160)(46 126 231 159)(47 125 232 158)(48 124 233 157)(49 123 234 156)(50 122 235 155)(51 121 236 154)(52 140 237 153)(53 139 238 152)(54 138 239 151)(55 137 240 150)(56 136 221 149)(57 135 222 148)(58 134 223 147)(59 133 224 146)(60 132 225 145)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222344444445566610···10101010101212121212121515202020202020202030···3060···60
size1111666024101012203030222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim111111112222222222244444444
type++++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10D20C4○D12S3×D4Q83S3S3×D5D42D5C2×S3×D5D12⋊D5S3×D20Dic3.D10
kernelD6.D20D304C4C6.Dic10C3×C4⋊Dic5C5×D6⋊C4D303C4C2×S3×Dic5C2×C5⋊D12C4⋊Dic5S3×C10D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3D6C10C10C10C2×C4C6C22C2C2C2
# reps111111111222142228411242444

Matrix representation of D6.D20 in GL4(𝔽61) generated by

1000
0100
0011
00600
,
1000
0100
0011
00060
,
73200
29200
005243
00189
,
532000
6800
003815
004623
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,60],[7,29,0,0,32,2,0,0,0,0,52,18,0,0,43,9],[53,6,0,0,20,8,0,0,0,0,38,46,0,0,15,23] >;

D6.D20 in GAP, Magma, Sage, TeX

D_6.D_{20}
% in TeX

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

G:=SmallGroup(480,503);
// by ID

G=gap.SmallGroup(480,503);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,58,1356,18822]);
// 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=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽