direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.D6, C30⋊2SD16, C60.83D4, Dic10⋊19D6, D12.36D10, C60.155C23, C15⋊5(C2×SD16), C6⋊2(D4.D5), (C2×D12).7D5, C30.83(C2×D4), (C2×C30).51D4, (C2×C20).94D6, (C6×Dic10)⋊9C2, (C2×Dic10)⋊7S3, (C10×D12).7C2, (C2×C12).94D10, C4.7(C15⋊D4), C15⋊3C8⋊39C22, C10⋊2(Q8⋊2S3), C12.30(C5⋊D4), C20.28(C3⋊D4), C20.94(C22×S3), C12.92(C22×D5), (C2×C60).189C22, (C5×D12).42C22, (C3×Dic10)⋊25C22, C22.21(C15⋊D4), C3⋊3(C2×D4.D5), C5⋊3(C2×Q8⋊2S3), C4.128(C2×S3×D5), (C2×C15⋊3C8)⋊17C2, C6.77(C2×C5⋊D4), (C2×C4).199(S3×D5), C10.78(C2×C3⋊D4), C2.11(C2×C15⋊D4), (C2×C6).54(C5⋊D4), (C2×C10).55(C3⋊D4), SmallGroup(480,384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.D6
G = < a,b,c,d | a2=b30=1, c4=d2=b15, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b19, dcd-1=c3 >
Subgroups: 572 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C2×SD16, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C2×C3⋊C8, Q8⋊2S3, C2×D12, C6×Q8, C3×Dic5, C60, S3×C10, C2×C30, C2×C5⋊2C8, D4.D5, C2×Dic10, D4×C10, C2×Q8⋊2S3, C15⋊3C8, C3×Dic10, C3×Dic10, C6×Dic5, C5×D12, C5×D12, C2×C60, S3×C2×C10, C2×D4.D5, C20.D6, C2×C15⋊3C8, C6×Dic10, C10×D12, C2×C20.D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, C2×SD16, C5⋊D4, C22×D5, Q8⋊2S3, C2×C3⋊D4, S3×D5, D4.D5, C2×C5⋊D4, C2×Q8⋊2S3, C15⋊D4, C2×S3×D5, C2×D4.D5, C20.D6, C2×C15⋊D4, C2×C20.D6
►On 240 points
Generators in S240
(1 86)(2 87)(3 88)(4 89)(5 90)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(127 196)(128 197)(129 198)(130 199)(131 200)(132 201)(133 202)(134 203)(135 204)(136 205)(137 206)(138 207)(139 208)(140 209)(141 210)(142 181)(143 182)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)(151 233)(152 234)(153 235)(154 236)(155 237)(156 238)(157 239)(158 240)(159 211)(160 212)(161 213)(162 214)(163 215)(164 216)(165 217)(166 218)(167 219)(168 220)(169 221)(170 222)(171 223)(172 224)(173 225)(174 226)(175 227)(176 228)(177 229)(178 230)(179 231)(180 232)
(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)
(1 157 36 127 16 172 51 142)(2 156 37 126 17 171 52 141)(3 155 38 125 18 170 53 140)(4 154 39 124 19 169 54 139)(5 153 40 123 20 168 55 138)(6 152 41 122 21 167 56 137)(7 151 42 121 22 166 57 136)(8 180 43 150 23 165 58 135)(9 179 44 149 24 164 59 134)(10 178 45 148 25 163 60 133)(11 177 46 147 26 162 31 132)(12 176 47 146 27 161 32 131)(13 175 48 145 28 160 33 130)(14 174 49 144 29 159 34 129)(15 173 50 143 30 158 35 128)(61 234 91 191 76 219 106 206)(62 233 92 190 77 218 107 205)(63 232 93 189 78 217 108 204)(64 231 94 188 79 216 109 203)(65 230 95 187 80 215 110 202)(66 229 96 186 81 214 111 201)(67 228 97 185 82 213 112 200)(68 227 98 184 83 212 113 199)(69 226 99 183 84 211 114 198)(70 225 100 182 85 240 115 197)(71 224 101 181 86 239 116 196)(72 223 102 210 87 238 117 195)(73 222 103 209 88 237 118 194)(74 221 104 208 89 236 119 193)(75 220 105 207 90 235 120 192)
(1 196 16 181)(2 185 17 200)(3 204 18 189)(4 193 19 208)(5 182 20 197)(6 201 21 186)(7 190 22 205)(8 209 23 194)(9 198 24 183)(10 187 25 202)(11 206 26 191)(12 195 27 210)(13 184 28 199)(14 203 29 188)(15 192 30 207)(31 234 46 219)(32 223 47 238)(33 212 48 227)(34 231 49 216)(35 220 50 235)(36 239 51 224)(37 228 52 213)(38 217 53 232)(39 236 54 221)(40 225 55 240)(41 214 56 229)(42 233 57 218)(43 222 58 237)(44 211 59 226)(45 230 60 215)(61 132 76 147)(62 121 77 136)(63 140 78 125)(64 129 79 144)(65 148 80 133)(66 137 81 122)(67 126 82 141)(68 145 83 130)(69 134 84 149)(70 123 85 138)(71 142 86 127)(72 131 87 146)(73 150 88 135)(74 139 89 124)(75 128 90 143)(91 162 106 177)(92 151 107 166)(93 170 108 155)(94 159 109 174)(95 178 110 163)(96 167 111 152)(97 156 112 171)(98 175 113 160)(99 164 114 179)(100 153 115 168)(101 172 116 157)(102 161 117 176)(103 180 118 165)(104 169 119 154)(105 158 120 173)
G:=sub<Sym(240)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,210)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,233)(152,234)(153,235)(154,236)(155,237)(156,238)(157,239)(158,240)(159,211)(160,212)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232), (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), (1,157,36,127,16,172,51,142)(2,156,37,126,17,171,52,141)(3,155,38,125,18,170,53,140)(4,154,39,124,19,169,54,139)(5,153,40,123,20,168,55,138)(6,152,41,122,21,167,56,137)(7,151,42,121,22,166,57,136)(8,180,43,150,23,165,58,135)(9,179,44,149,24,164,59,134)(10,178,45,148,25,163,60,133)(11,177,46,147,26,162,31,132)(12,176,47,146,27,161,32,131)(13,175,48,145,28,160,33,130)(14,174,49,144,29,159,34,129)(15,173,50,143,30,158,35,128)(61,234,91,191,76,219,106,206)(62,233,92,190,77,218,107,205)(63,232,93,189,78,217,108,204)(64,231,94,188,79,216,109,203)(65,230,95,187,80,215,110,202)(66,229,96,186,81,214,111,201)(67,228,97,185,82,213,112,200)(68,227,98,184,83,212,113,199)(69,226,99,183,84,211,114,198)(70,225,100,182,85,240,115,197)(71,224,101,181,86,239,116,196)(72,223,102,210,87,238,117,195)(73,222,103,209,88,237,118,194)(74,221,104,208,89,236,119,193)(75,220,105,207,90,235,120,192), (1,196,16,181)(2,185,17,200)(3,204,18,189)(4,193,19,208)(5,182,20,197)(6,201,21,186)(7,190,22,205)(8,209,23,194)(9,198,24,183)(10,187,25,202)(11,206,26,191)(12,195,27,210)(13,184,28,199)(14,203,29,188)(15,192,30,207)(31,234,46,219)(32,223,47,238)(33,212,48,227)(34,231,49,216)(35,220,50,235)(36,239,51,224)(37,228,52,213)(38,217,53,232)(39,236,54,221)(40,225,55,240)(41,214,56,229)(42,233,57,218)(43,222,58,237)(44,211,59,226)(45,230,60,215)(61,132,76,147)(62,121,77,136)(63,140,78,125)(64,129,79,144)(65,148,80,133)(66,137,81,122)(67,126,82,141)(68,145,83,130)(69,134,84,149)(70,123,85,138)(71,142,86,127)(72,131,87,146)(73,150,88,135)(74,139,89,124)(75,128,90,143)(91,162,106,177)(92,151,107,166)(93,170,108,155)(94,159,109,174)(95,178,110,163)(96,167,111,152)(97,156,112,171)(98,175,113,160)(99,164,114,179)(100,153,115,168)(101,172,116,157)(102,161,117,176)(103,180,118,165)(104,169,119,154)(105,158,120,173)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(141,210)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,233)(152,234)(153,235)(154,236)(155,237)(156,238)(157,239)(158,240)(159,211)(160,212)(161,213)(162,214)(163,215)(164,216)(165,217)(166,218)(167,219)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232), (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), (1,157,36,127,16,172,51,142)(2,156,37,126,17,171,52,141)(3,155,38,125,18,170,53,140)(4,154,39,124,19,169,54,139)(5,153,40,123,20,168,55,138)(6,152,41,122,21,167,56,137)(7,151,42,121,22,166,57,136)(8,180,43,150,23,165,58,135)(9,179,44,149,24,164,59,134)(10,178,45,148,25,163,60,133)(11,177,46,147,26,162,31,132)(12,176,47,146,27,161,32,131)(13,175,48,145,28,160,33,130)(14,174,49,144,29,159,34,129)(15,173,50,143,30,158,35,128)(61,234,91,191,76,219,106,206)(62,233,92,190,77,218,107,205)(63,232,93,189,78,217,108,204)(64,231,94,188,79,216,109,203)(65,230,95,187,80,215,110,202)(66,229,96,186,81,214,111,201)(67,228,97,185,82,213,112,200)(68,227,98,184,83,212,113,199)(69,226,99,183,84,211,114,198)(70,225,100,182,85,240,115,197)(71,224,101,181,86,239,116,196)(72,223,102,210,87,238,117,195)(73,222,103,209,88,237,118,194)(74,221,104,208,89,236,119,193)(75,220,105,207,90,235,120,192), (1,196,16,181)(2,185,17,200)(3,204,18,189)(4,193,19,208)(5,182,20,197)(6,201,21,186)(7,190,22,205)(8,209,23,194)(9,198,24,183)(10,187,25,202)(11,206,26,191)(12,195,27,210)(13,184,28,199)(14,203,29,188)(15,192,30,207)(31,234,46,219)(32,223,47,238)(33,212,48,227)(34,231,49,216)(35,220,50,235)(36,239,51,224)(37,228,52,213)(38,217,53,232)(39,236,54,221)(40,225,55,240)(41,214,56,229)(42,233,57,218)(43,222,58,237)(44,211,59,226)(45,230,60,215)(61,132,76,147)(62,121,77,136)(63,140,78,125)(64,129,79,144)(65,148,80,133)(66,137,81,122)(67,126,82,141)(68,145,83,130)(69,134,84,149)(70,123,85,138)(71,142,86,127)(72,131,87,146)(73,150,88,135)(74,139,89,124)(75,128,90,143)(91,162,106,177)(92,151,107,166)(93,170,108,155)(94,159,109,174)(95,178,110,163)(96,167,111,152)(97,156,112,171)(98,175,113,160)(99,164,114,179)(100,153,115,168)(101,172,116,157)(102,161,117,176)(103,180,118,165)(104,169,119,154)(105,158,120,173) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(121,190),(122,191),(123,192),(124,193),(125,194),(126,195),(127,196),(128,197),(129,198),(130,199),(131,200),(132,201),(133,202),(134,203),(135,204),(136,205),(137,206),(138,207),(139,208),(140,209),(141,210),(142,181),(143,182),(144,183),(145,184),(146,185),(147,186),(148,187),(149,188),(150,189),(151,233),(152,234),(153,235),(154,236),(155,237),(156,238),(157,239),(158,240),(159,211),(160,212),(161,213),(162,214),(163,215),(164,216),(165,217),(166,218),(167,219),(168,220),(169,221),(170,222),(171,223),(172,224),(173,225),(174,226),(175,227),(176,228),(177,229),(178,230),(179,231),(180,232)], [(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)], [(1,157,36,127,16,172,51,142),(2,156,37,126,17,171,52,141),(3,155,38,125,18,170,53,140),(4,154,39,124,19,169,54,139),(5,153,40,123,20,168,55,138),(6,152,41,122,21,167,56,137),(7,151,42,121,22,166,57,136),(8,180,43,150,23,165,58,135),(9,179,44,149,24,164,59,134),(10,178,45,148,25,163,60,133),(11,177,46,147,26,162,31,132),(12,176,47,146,27,161,32,131),(13,175,48,145,28,160,33,130),(14,174,49,144,29,159,34,129),(15,173,50,143,30,158,35,128),(61,234,91,191,76,219,106,206),(62,233,92,190,77,218,107,205),(63,232,93,189,78,217,108,204),(64,231,94,188,79,216,109,203),(65,230,95,187,80,215,110,202),(66,229,96,186,81,214,111,201),(67,228,97,185,82,213,112,200),(68,227,98,184,83,212,113,199),(69,226,99,183,84,211,114,198),(70,225,100,182,85,240,115,197),(71,224,101,181,86,239,116,196),(72,223,102,210,87,238,117,195),(73,222,103,209,88,237,118,194),(74,221,104,208,89,236,119,193),(75,220,105,207,90,235,120,192)], [(1,196,16,181),(2,185,17,200),(3,204,18,189),(4,193,19,208),(5,182,20,197),(6,201,21,186),(7,190,22,205),(8,209,23,194),(9,198,24,183),(10,187,25,202),(11,206,26,191),(12,195,27,210),(13,184,28,199),(14,203,29,188),(15,192,30,207),(31,234,46,219),(32,223,47,238),(33,212,48,227),(34,231,49,216),(35,220,50,235),(36,239,51,224),(37,228,52,213),(38,217,53,232),(39,236,54,221),(40,225,55,240),(41,214,56,229),(42,233,57,218),(43,222,58,237),(44,211,59,226),(45,230,60,215),(61,132,76,147),(62,121,77,136),(63,140,78,125),(64,129,79,144),(65,148,80,133),(66,137,81,122),(67,126,82,141),(68,145,83,130),(69,134,84,149),(70,123,85,138),(71,142,86,127),(72,131,87,146),(73,150,88,135),(74,139,89,124),(75,128,90,143),(91,162,106,177),(92,151,107,166),(93,170,108,155),(94,159,109,174),(95,178,110,163),(96,167,111,152),(97,156,112,171),(98,175,113,160),(99,164,114,179),(100,153,115,168),(101,172,116,157),(102,161,117,176),(103,180,118,165),(104,169,119,154),(105,158,120,173)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | SD16 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | Q8⋊2S3 | S3×D5 | D4.D5 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | C20.D6 |
| kernel | C2×C20.D6 | C20.D6 | C2×C15⋊3C8 | C6×Dic10 | C10×D12 | C2×Dic10 | C60 | C2×C30 | C2×D12 | Dic10 | C2×C20 | C30 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C20.D6 ►in GL6(𝔽241)
| 1 | 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 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 87 | 0 | 0 | 0 | 0 | 0 |
| 191 | 205 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 239 | 54 |
| 0 | 0 | 0 | 0 | 174 | 1 |
| 119 | 18 | 0 | 0 | 0 | 0 |
| 204 | 122 | 0 | 0 | 0 | 0 |
| 0 | 0 | 236 | 13 | 0 | 0 |
| 0 | 0 | 124 | 208 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 54 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 119 | 18 | 0 | 0 | 0 | 0 |
| 97 | 122 | 0 | 0 | 0 | 0 |
| 0 | 0 | 208 | 36 | 0 | 0 |
| 0 | 0 | 117 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[87,191,0,0,0,0,0,205,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,239,174,0,0,0,0,54,1],[119,204,0,0,0,0,18,122,0,0,0,0,0,0,236,124,0,0,0,0,13,208,0,0,0,0,0,0,240,0,0,0,0,0,54,1],[119,97,0,0,0,0,18,122,0,0,0,0,0,0,208,117,0,0,0,0,36,33,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
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(480,384);
// by ID
G=gap.SmallGroup(480,384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=1,c^4=d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^19,d*c*d^-1=c^3>;
// generators/relations