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