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G = (C2×C6)⋊D20order 480 = 25·3·5

1st semidirect product of C2×C6 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C6)⋊1D20, (C2×C30)⋊8D4, C34(C207D4), C6.70(C2×D20), C1523(C4⋊D4), (C5×Dic3)⋊16D4, C30.249(C2×D4), C10.164(S3×D4), D304C435C2, C23.33(S3×D5), C6.87(C4○D20), Dic35(C5⋊D4), C54(C23.14D6), C222(C3⋊D20), C6.Dic1038C2, (C2×Dic5).65D6, (C22×Dic3)⋊7D5, (C22×D5).32D6, (C22×C6).45D10, C30.158(C4○D4), D10⋊Dic336C2, (C2×C30).211C23, (C22×C10).111D6, C10.59(D42S3), (C2×Dic3).166D10, (C22×C30).73C22, C2.30(Dic5.D6), (C6×Dic5).122C22, (C22×D15).70C22, (C2×Dic15).144C22, (C10×Dic3).204C22, (C2×C5⋊D4)⋊7S3, (C6×C5⋊D4)⋊7C2, (Dic3×C2×C10)⋊7C2, C6.68(C2×C5⋊D4), C2.44(S3×C5⋊D4), (C2×C3⋊D20)⋊15C2, (C2×C157D4)⋊19C2, C10.23(C2×C3⋊D4), C2.25(C2×C3⋊D20), (D5×C2×C6).55C22, (C2×C10)⋊12(C3⋊D4), C22.240(C2×S3×D5), (C2×C6).223(C22×D5), (C2×C10).223(C22×S3), SmallGroup(480,645)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C6)⋊D20
C1C5C15C30C2×C30D5×C2×C6C2×C3⋊D20 — (C2×C6)⋊D20
C15C2×C30 — (C2×C6)⋊D20
C1C22C23

Generators and relations for (C2×C6)⋊D20
 G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, ac=ca, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1052 in 188 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], C5, S3, C6 [×3], C6 [×3], C2×C4 [×6], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×3], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6, C22×C6, C3×D5, D15, C30 [×3], C30 [×2], C4⋊D4, D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20 [×4], C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15, C6×D5 [×3], D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23.14D6, C3⋊D20 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], D5×C2×C6, C22×D15, C22×C30, C207D4, D10⋊Dic3, D304C4, C6.Dic10, C2×C3⋊D20, C6×C5⋊D4, Dic3×C2×C10, C2×C157D4, (C2×C6)⋊D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, C23.14D6, C3⋊D20 [×2], C2×S3×D5, C207D4, Dic5.D6, C2×C3⋊D20, S3×C5⋊D4, (C2×C6)⋊D20

Smallest permutation representation of (C2×C6)⋊D20
On 240 points
Generators in S240
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 121)(58 122)(59 123)(60 124)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(141 197)(142 198)(143 199)(144 200)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(151 187)(152 188)(153 189)(154 190)(155 191)(156 192)(157 193)(158 194)(159 195)(160 196)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(201 233)(202 234)(203 235)(204 236)(205 237)(206 238)(207 239)(208 240)(209 221)(210 222)(211 223)(212 224)(213 225)(214 226)(215 227)(216 228)(217 229)(218 230)(219 231)(220 232)
(1 110 64 172 36 99)(2 100 37 173 65 111)(3 112 66 174 38 81)(4 82 39 175 67 113)(5 114 68 176 40 83)(6 84 21 177 69 115)(7 116 70 178 22 85)(8 86 23 179 71 117)(9 118 72 180 24 87)(10 88 25 161 73 119)(11 120 74 162 26 89)(12 90 27 163 75 101)(13 102 76 164 28 91)(14 92 29 165 77 103)(15 104 78 166 30 93)(16 94 31 167 79 105)(17 106 80 168 32 95)(18 96 33 169 61 107)(19 108 62 170 34 97)(20 98 35 171 63 109)(41 231 195 135 209 149)(42 150 210 136 196 232)(43 233 197 137 211 151)(44 152 212 138 198 234)(45 235 199 139 213 153)(46 154 214 140 200 236)(47 237 181 121 215 155)(48 156 216 122 182 238)(49 239 183 123 217 157)(50 158 218 124 184 240)(51 221 185 125 219 159)(52 160 220 126 186 222)(53 223 187 127 201 141)(54 142 202 128 188 224)(55 225 189 129 203 143)(56 144 204 130 190 226)(57 227 191 131 205 145)(58 146 206 132 192 228)(59 229 193 133 207 147)(60 148 208 134 194 230)
(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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 239)(22 238)(23 237)(24 236)(25 235)(26 234)(27 233)(28 232)(29 231)(30 230)(31 229)(32 228)(33 227)(34 226)(35 225)(36 224)(37 223)(38 222)(39 221)(40 240)(41 165)(42 164)(43 163)(44 162)(45 161)(46 180)(47 179)(48 178)(49 177)(50 176)(51 175)(52 174)(53 173)(54 172)(55 171)(56 170)(57 169)(58 168)(59 167)(60 166)(61 145)(62 144)(63 143)(64 142)(65 141)(66 160)(67 159)(68 158)(69 157)(70 156)(71 155)(72 154)(73 153)(74 152)(75 151)(76 150)(77 149)(78 148)(79 147)(80 146)(81 186)(82 185)(83 184)(84 183)(85 182)(86 181)(87 200)(88 199)(89 198)(90 197)(91 196)(92 195)(93 194)(94 193)(95 192)(96 191)(97 190)(98 189)(99 188)(100 187)(101 211)(102 210)(103 209)(104 208)(105 207)(106 206)(107 205)(108 204)(109 203)(110 202)(111 201)(112 220)(113 219)(114 218)(115 217)(116 216)(117 215)(118 214)(119 213)(120 212)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10N12A12B15A15B20A···20P30A···30N
order12222222344444455666666610···101212151520···2030···30
size1111222060266662060222224420202···22020446···64···4

72 irreducible representations

dim11111111222222222222224444444
type+++++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10C3⋊D4C5⋊D4D20C4○D20S3×D4D42S3S3×D5C3⋊D20C2×S3×D5Dic5.D6S3×C5⋊D4
kernel(C2×C6)⋊D20D10⋊Dic3D304C4C6.Dic10C2×C3⋊D20C6×C5⋊D4Dic3×C2×C10C2×C157D4C2×C5⋊D4C5×Dic3C2×C30C22×Dic3C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6C2×C10Dic3C2×C6C6C10C10C23C22C22C2C2
# reps11111111122211124248881124244

Matrix representation of (C2×C6)⋊D20 in GL4(𝔽61) generated by

603600
0100
00600
00060
,
60000
06000
00119
004859
,
382600
05300
002830
004533
,
331100
122800
001943
002042
G:=sub<GL(4,GF(61))| [60,0,0,0,36,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,48,0,0,19,59],[38,0,0,0,26,53,0,0,0,0,28,45,0,0,30,33],[33,12,0,0,11,28,0,0,0,0,19,20,0,0,43,42] >;

(C2×C6)⋊D20 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes D_{20}
% in TeX

G:=Group("(C2xC6):D20");
// GroupNames label

G:=SmallGroup(480,645);
// by ID

G=gap.SmallGroup(480,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽