direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D60⋊C2, C30.9C24, Dic10⋊24D6, D60⋊31C22, D30.3C23, C60.113C23, (C4×S3)⋊15D10, (C2×D60)⋊23C2, C30⋊4(C4○D4), C6⋊1(C4○D20), C6.9(C23×D5), (C6×Dic10)⋊8C2, (C2×C20).307D6, C5⋊D12⋊9C22, C10.9(S3×C23), (S3×C20)⋊17C22, C10⋊1(Q8⋊3S3), (C2×Dic10)⋊15S3, (C2×C12).166D10, D30.C2⋊5C22, D6.23(C22×D5), (S3×C10).26C23, (C2×C60).126C22, C20.162(C22×S3), (C2×C30).228C23, (C2×Dic5).137D6, (C22×S3).81D10, C12.127(C22×D5), Dic5.6(C22×S3), (C3×Dic5).6C23, (C2×Dic3).190D10, (C3×Dic10)⋊21C22, (C5×Dic3).27C23, Dic3.33(C22×D5), (C6×Dic5).129C22, (C22×D15).72C22, (C10×Dic3).208C22, (S3×C2×C4)⋊4D5, (S3×C2×C20)⋊5C2, C3⋊1(C2×C4○D20), C15⋊4(C2×C4○D4), C5⋊1(C2×Q8⋊3S3), C4.111(C2×S3×D5), (C2×C5⋊D12)⋊18C2, C2.13(C22×S3×D5), C22.97(C2×S3×D5), (C2×C4).117(S3×D5), (C2×D30.C2)⋊19C2, (S3×C2×C10).99C22, (C2×C6).238(C22×D5), (C2×C10).238(C22×S3), SmallGroup(480,1081)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D60⋊C2
G = < a,b,c,d | a2=b60=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b41, dcd=b10c >
Subgroups: 1660 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×S3, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C5×S3, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, S3×C2×C4, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10, D30, D30, C2×C30, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×Q8⋊3S3, D30.C2, C5⋊D12, C3×Dic10, C6×Dic5, S3×C20, C10×Dic3, D60, C2×C60, S3×C2×C10, C22×D15, C2×C4○D20, D60⋊C2, C2×D30.C2, C2×C5⋊D12, C6×Dic10, S3×C2×C20, C2×D60, C2×D60⋊C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, Q8⋊3S3, S3×C23, S3×D5, C4○D20, C23×D5, C2×Q8⋊3S3, C2×S3×D5, C2×C4○D20, D60⋊C2, C22×S3×D5, C2×D60⋊C2
►On 240 points
Generators in S240
(1 117)(2 118)(3 119)(4 120)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 181)(167 182)(168 183)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 120)(58 119)(59 118)(60 117)(121 213)(122 212)(123 211)(124 210)(125 209)(126 208)(127 207)(128 206)(129 205)(130 204)(131 203)(132 202)(133 201)(134 200)(135 199)(136 198)(137 197)(138 196)(139 195)(140 194)(141 193)(142 192)(143 191)(144 190)(145 189)(146 188)(147 187)(148 186)(149 185)(150 184)(151 183)(152 182)(153 181)(154 240)(155 239)(156 238)(157 237)(158 236)(159 235)(160 234)(161 233)(162 232)(163 231)(164 230)(165 229)(166 228)(167 227)(168 226)(169 225)(170 224)(171 223)(172 222)(173 221)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)
(1 175)(2 156)(3 137)(4 178)(5 159)(6 140)(7 121)(8 162)(9 143)(10 124)(11 165)(12 146)(13 127)(14 168)(15 149)(16 130)(17 171)(18 152)(19 133)(20 174)(21 155)(22 136)(23 177)(24 158)(25 139)(26 180)(27 161)(28 142)(29 123)(30 164)(31 145)(32 126)(33 167)(34 148)(35 129)(36 170)(37 151)(38 132)(39 173)(40 154)(41 135)(42 176)(43 157)(44 138)(45 179)(46 160)(47 141)(48 122)(49 163)(50 144)(51 125)(52 166)(53 147)(54 128)(55 169)(56 150)(57 131)(58 172)(59 153)(60 134)(61 234)(62 215)(63 196)(64 237)(65 218)(66 199)(67 240)(68 221)(69 202)(70 183)(71 224)(72 205)(73 186)(74 227)(75 208)(76 189)(77 230)(78 211)(79 192)(80 233)(81 214)(82 195)(83 236)(84 217)(85 198)(86 239)(87 220)(88 201)(89 182)(90 223)(91 204)(92 185)(93 226)(94 207)(95 188)(96 229)(97 210)(98 191)(99 232)(100 213)(101 194)(102 235)(103 216)(104 197)(105 238)(106 219)(107 200)(108 181)(109 222)(110 203)(111 184)(112 225)(113 206)(114 187)(115 228)(116 209)(117 190)(118 231)(119 212)(120 193)
G:=sub<Sym(240)| (1,117)(2,118)(3,119)(4,120)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,120)(58,119)(59,118)(60,117)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,183)(152,182)(153,181)(154,240)(155,239)(156,238)(157,237)(158,236)(159,235)(160,234)(161,233)(162,232)(163,231)(164,230)(165,229)(166,228)(167,227)(168,226)(169,225)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214), (1,175)(2,156)(3,137)(4,178)(5,159)(6,140)(7,121)(8,162)(9,143)(10,124)(11,165)(12,146)(13,127)(14,168)(15,149)(16,130)(17,171)(18,152)(19,133)(20,174)(21,155)(22,136)(23,177)(24,158)(25,139)(26,180)(27,161)(28,142)(29,123)(30,164)(31,145)(32,126)(33,167)(34,148)(35,129)(36,170)(37,151)(38,132)(39,173)(40,154)(41,135)(42,176)(43,157)(44,138)(45,179)(46,160)(47,141)(48,122)(49,163)(50,144)(51,125)(52,166)(53,147)(54,128)(55,169)(56,150)(57,131)(58,172)(59,153)(60,134)(61,234)(62,215)(63,196)(64,237)(65,218)(66,199)(67,240)(68,221)(69,202)(70,183)(71,224)(72,205)(73,186)(74,227)(75,208)(76,189)(77,230)(78,211)(79,192)(80,233)(81,214)(82,195)(83,236)(84,217)(85,198)(86,239)(87,220)(88,201)(89,182)(90,223)(91,204)(92,185)(93,226)(94,207)(95,188)(96,229)(97,210)(98,191)(99,232)(100,213)(101,194)(102,235)(103,216)(104,197)(105,238)(106,219)(107,200)(108,181)(109,222)(110,203)(111,184)(112,225)(113,206)(114,187)(115,228)(116,209)(117,190)(118,231)(119,212)(120,193)>;
G:=Group( (1,117)(2,118)(3,119)(4,120)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,120)(58,119)(59,118)(60,117)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,183)(152,182)(153,181)(154,240)(155,239)(156,238)(157,237)(158,236)(159,235)(160,234)(161,233)(162,232)(163,231)(164,230)(165,229)(166,228)(167,227)(168,226)(169,225)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214), (1,175)(2,156)(3,137)(4,178)(5,159)(6,140)(7,121)(8,162)(9,143)(10,124)(11,165)(12,146)(13,127)(14,168)(15,149)(16,130)(17,171)(18,152)(19,133)(20,174)(21,155)(22,136)(23,177)(24,158)(25,139)(26,180)(27,161)(28,142)(29,123)(30,164)(31,145)(32,126)(33,167)(34,148)(35,129)(36,170)(37,151)(38,132)(39,173)(40,154)(41,135)(42,176)(43,157)(44,138)(45,179)(46,160)(47,141)(48,122)(49,163)(50,144)(51,125)(52,166)(53,147)(54,128)(55,169)(56,150)(57,131)(58,172)(59,153)(60,134)(61,234)(62,215)(63,196)(64,237)(65,218)(66,199)(67,240)(68,221)(69,202)(70,183)(71,224)(72,205)(73,186)(74,227)(75,208)(76,189)(77,230)(78,211)(79,192)(80,233)(81,214)(82,195)(83,236)(84,217)(85,198)(86,239)(87,220)(88,201)(89,182)(90,223)(91,204)(92,185)(93,226)(94,207)(95,188)(96,229)(97,210)(98,191)(99,232)(100,213)(101,194)(102,235)(103,216)(104,197)(105,238)(106,219)(107,200)(108,181)(109,222)(110,203)(111,184)(112,225)(113,206)(114,187)(115,228)(116,209)(117,190)(118,231)(119,212)(120,193) );
G=PermutationGroup([[(1,117),(2,118),(3,119),(4,120),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,226),(152,227),(153,228),(154,229),(155,230),(156,231),(157,232),(158,233),(159,234),(160,235),(161,236),(162,237),(163,238),(164,239),(165,240),(166,181),(167,182),(168,183),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195)], [(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)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,84),(34,83),(35,82),(36,81),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,120),(58,119),(59,118),(60,117),(121,213),(122,212),(123,211),(124,210),(125,209),(126,208),(127,207),(128,206),(129,205),(130,204),(131,203),(132,202),(133,201),(134,200),(135,199),(136,198),(137,197),(138,196),(139,195),(140,194),(141,193),(142,192),(143,191),(144,190),(145,189),(146,188),(147,187),(148,186),(149,185),(150,184),(151,183),(152,182),(153,181),(154,240),(155,239),(156,238),(157,237),(158,236),(159,235),(160,234),(161,233),(162,232),(163,231),(164,230),(165,229),(166,228),(167,227),(168,226),(169,225),(170,224),(171,223),(172,222),(173,221),(174,220),(175,219),(176,218),(177,217),(178,216),(179,215),(180,214)], [(1,175),(2,156),(3,137),(4,178),(5,159),(6,140),(7,121),(8,162),(9,143),(10,124),(11,165),(12,146),(13,127),(14,168),(15,149),(16,130),(17,171),(18,152),(19,133),(20,174),(21,155),(22,136),(23,177),(24,158),(25,139),(26,180),(27,161),(28,142),(29,123),(30,164),(31,145),(32,126),(33,167),(34,148),(35,129),(36,170),(37,151),(38,132),(39,173),(40,154),(41,135),(42,176),(43,157),(44,138),(45,179),(46,160),(47,141),(48,122),(49,163),(50,144),(51,125),(52,166),(53,147),(54,128),(55,169),(56,150),(57,131),(58,172),(59,153),(60,134),(61,234),(62,215),(63,196),(64,237),(65,218),(66,199),(67,240),(68,221),(69,202),(70,183),(71,224),(72,205),(73,186),(74,227),(75,208),(76,189),(77,230),(78,211),(79,192),(80,233),(81,214),(82,195),(83,236),(84,217),(85,198),(86,239),(87,220),(88,201),(89,182),(90,223),(91,204),(92,185),(93,226),(94,207),(95,188),(96,229),(97,210),(98,191),(99,232),(100,213),(101,194),(102,235),(103,216),(104,197),(105,238),(106,219),(107,200),(108,181),(109,222),(110,203),(111,184),(112,225),(113,206),(114,187),(115,228),(116,209),(117,190),(118,231),(119,212),(120,193)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 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 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | Q8⋊3S3 | S3×D5 | C2×S3×D5 | C2×S3×D5 | D60⋊C2 |
| kernel | C2×D60⋊C2 | D60⋊C2 | C2×D30.C2 | C2×C5⋊D12 | C6×Dic10 | S3×C2×C20 | C2×D60 | C2×Dic10 | S3×C2×C4 | Dic10 | C2×Dic5 | C2×C20 | C30 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 4 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C2×D60⋊C2 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 27 | 34 | 0 | 0 |
| 27 | 25 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 |
| 27 | 34 | 0 | 0 |
| 36 | 34 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
| 31 | 17 | 0 | 0 |
| 44 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[27,27,0,0,34,25,0,0,0,0,1,1,0,0,60,0],[27,36,0,0,34,34,0,0,0,0,60,60,0,0,0,1],[31,44,0,0,17,30,0,0,0,0,0,1,0,0,1,0] >;
C2×D60⋊C2 in GAP, Magma, Sage, TeX
C_2\times D_{60}\rtimes C_2 % in TeX
G:=Group("C2xD60:C2"); // GroupNames label
G:=SmallGroup(480,1081);
// by ID
G=gap.SmallGroup(480,1081);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^60=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^41,d*c*d=b^10*c>;
// generators/relations