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G = (C2×C20).D6order 480 = 25·3·5

1st non-split extension by C2×C20 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C415D5, (C2×C20).1D6, D6⋊Dic51C2, C6.123(D4×D5), C30.98(C2×D4), D304C41C2, (C2×Dic10)⋊2S3, C6.1(C4○D20), C30.1(C4○D4), C151(C4.4D4), (C3×Dic5).5D4, D303C413C2, (Dic3×Dic5)⋊1C2, (C6×Dic10)⋊14C2, C6.1(D42D5), (C2×C12).215D10, C52(C12.23D4), (C2×C30).16C23, (C2×Dic5).80D6, (C22×S3).1D10, C2.6(D60⋊C2), C2.6(D12⋊D5), (C2×C60).246C22, C10.1(Q83S3), (C2×Dic3).72D10, Dic5.9(C3⋊D4), C36(Dic5.5D4), (C6×Dic5).3C22, (C10×Dic3).3C22, (C2×Dic15).24C22, (C22×D15).15C22, C2.8(D5×C3⋊D4), (C5×D6⋊C4)⋊15C2, (C2×C4).19(S3×D5), (C2×C5⋊D12).1C2, C10.25(C2×C3⋊D4), (S3×C2×C10).1C22, C22.113(C2×S3×D5), (C2×C6).28(C22×D5), (C2×C10).28(C22×S3), SmallGroup(480,402)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C20).D6
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — (C2×C20).D6
C15C2×C30 — (C2×C20).D6
C1C22C2×C4

Generators and relations for (C2×C20).D6
 G = < a,b,c,d | a2=b20=1, c6=b10, d2=a, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=c5 >

Subgroups: 860 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5, C10 [×3], C10, Dic3 [×2], C12 [×4], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], D12 [×2], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×S3, C5×S3, D15, C30 [×3], C4.4D4, Dic10 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4 [×3], C2×D12, C6×Q8, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15, C60, S3×C10 [×3], D30 [×3], C2×C30, C4×Dic5, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C12.23D4, C5⋊D12 [×2], C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic5.5D4, Dic3×Dic5, D6⋊Dic5, D304C4, C5×D6⋊C4, D303C4, C2×C5⋊D12, C6×Dic10, (C2×C20).D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4.4D4, C22×D5, Q83S3 [×2], C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D42D5, C12.23D4, C2×S3×D5, Dic5.5D4, D12⋊D5, D60⋊C2, D5×C3⋊D4, (C2×C20).D6

Smallest permutation representation of (C2×C20).D6
On 240 points
Generators in S240
(1 165)(2 166)(3 167)(4 168)(5 169)(6 170)(7 171)(8 172)(9 173)(10 174)(11 175)(12 176)(13 177)(14 178)(15 179)(16 180)(17 161)(18 162)(19 163)(20 164)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 159)(34 160)(35 141)(36 142)(37 143)(38 144)(39 145)(40 146)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 201)(91 202)(92 203)(93 204)(94 205)(95 206)(96 207)(97 208)(98 209)(99 210)(100 211)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 197)(113 198)(114 199)(115 200)(116 181)(117 182)(118 183)(119 184)(120 185)(121 223)(122 224)(123 225)(124 226)(125 227)(126 228)(127 229)(128 230)(129 231)(130 232)(131 233)(132 234)(133 235)(134 236)(135 237)(136 238)(137 239)(138 240)(139 221)(140 222)
(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)
(1 126 84 101 56 153 11 136 94 111 46 143)(2 125 85 120 57 152 12 135 95 110 47 142)(3 124 86 119 58 151 13 134 96 109 48 141)(4 123 87 118 59 150 14 133 97 108 49 160)(5 122 88 117 60 149 15 132 98 107 50 159)(6 121 89 116 41 148 16 131 99 106 51 158)(7 140 90 115 42 147 17 130 100 105 52 157)(8 139 91 114 43 146 18 129 81 104 53 156)(9 138 92 113 44 145 19 128 82 103 54 155)(10 137 93 112 45 144 20 127 83 102 55 154)(21 161 232 211 190 66 31 171 222 201 200 76)(22 180 233 210 191 65 32 170 223 220 181 75)(23 179 234 209 192 64 33 169 224 219 182 74)(24 178 235 208 193 63 34 168 225 218 183 73)(25 177 236 207 194 62 35 167 226 217 184 72)(26 176 237 206 195 61 36 166 227 216 185 71)(27 175 238 205 196 80 37 165 228 215 186 70)(28 174 239 204 197 79 38 164 229 214 187 69)(29 173 240 203 198 78 39 163 230 213 188 68)(30 172 221 202 199 77 40 162 231 212 189 67)
(1 89 165 220)(2 219 166 88)(3 87 167 218)(4 217 168 86)(5 85 169 216)(6 215 170 84)(7 83 171 214)(8 213 172 82)(9 81 173 212)(10 211 174 100)(11 99 175 210)(12 209 176 98)(13 97 177 208)(14 207 178 96)(15 95 179 206)(16 205 180 94)(17 93 161 204)(18 203 162 92)(19 91 163 202)(20 201 164 90)(21 187 147 102)(22 101 148 186)(23 185 149 120)(24 119 150 184)(25 183 151 118)(26 117 152 182)(27 181 153 116)(28 115 154 200)(29 199 155 114)(30 113 156 198)(31 197 157 112)(32 111 158 196)(33 195 159 110)(34 109 160 194)(35 193 141 108)(36 107 142 192)(37 191 143 106)(38 105 144 190)(39 189 145 104)(40 103 146 188)(41 80 75 46)(42 45 76 79)(43 78 77 44)(47 74 61 60)(48 59 62 73)(49 72 63 58)(50 57 64 71)(51 70 65 56)(52 55 66 69)(53 68 67 54)(121 238 223 136)(122 135 224 237)(123 236 225 134)(124 133 226 235)(125 234 227 132)(126 131 228 233)(127 232 229 130)(128 129 230 231)(137 222 239 140)(138 139 240 221)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222223444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111126024661010203030222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C3⋊D4C4○D20Q83S3S3×D5D4×D5D42D5C2×S3×D5D12⋊D5D60⋊C2D5×C3⋊D4
kernel(C2×C20).D6Dic3×Dic5D6⋊Dic5D304C4C5×D6⋊C4D303C4C2×C5⋊D12C6×Dic10C2×Dic10C3×Dic5D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5C6C10C2×C4C6C6C22C2C2C2
# reps111111111222142224822222444

Matrix representation of (C2×C20).D6 in GL6(𝔽61)

6000000
0600000
001000
000100
000010
000001
,
50600000
0110000
00573600
00253400
000010
000001
,
1500000
39600000
00535300
0031800
000001
0000601
,
1100000
0110000
00343400
00362700
0000160
0000060

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,0,0,0,0,0,60,11,0,0,0,0,0,0,57,25,0,0,0,0,36,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,39,0,0,0,0,50,60,0,0,0,0,0,0,53,31,0,0,0,0,53,8,0,0,0,0,0,0,0,60,0,0,0,0,1,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,34,36,0,0,0,0,34,27,0,0,0,0,0,0,1,0,0,0,0,0,60,60] >;

(C2×C20).D6 in GAP, Magma, Sage, TeX

(C_2\times C_{20}).D_6
% in TeX

G:=Group("(C2xC20).D6");
// GroupNames label

G:=SmallGroup(480,402);
// by ID

G=gap.SmallGroup(480,402);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,254,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^6=b^10,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=c^5>;
// generators/relations

׿
×
𝔽