direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D60⋊11C2, D60⋊42C22, C30.57C24, C23.32D30, C60.251C23, D30.24C23, Dic30⋊38C22, Dic15.27C23, (C2×C4)⋊10D30, (C2×C20)⋊34D6, (C2×D60)⋊30C2, C6⋊5(C4○D20), (C2×C12)⋊34D10, (C22×C4)⋊8D15, (C22×C12)⋊8D5, C30⋊11(C4○D4), C10⋊5(C4○D12), (C22×C20)⋊12S3, (C22×C60)⋊12C2, (C2×C60)⋊45C22, C2.5(C23×D15), C6.57(C23×D5), (C2×Dic30)⋊31C2, (C4×D15)⋊20C22, C15⋊7D4⋊22C22, C10.57(S3×C23), C4.42(C22×D15), (C2×C30).321C23, C20.230(C22×S3), (C22×C6).127D10, C12.232(C22×D5), (C22×C10).145D6, C22.5(C22×D15), (C22×C30).150C22, (C22×D15).90C22, (C2×Dic15).177C22, C3⋊6(C2×C4○D20), C5⋊6(C2×C4○D12), (C2×C4×D15)⋊21C2, C15⋊20(C2×C4○D4), (C2×C15⋊7D4)⋊27C2, (C2×C6).317(C22×D5), (C2×C10).316(C22×S3), SmallGroup(480,1168)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1716 in 328 conjugacy classes, 127 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×4], D6 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, D15 [×4], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, Dic15 [×4], C60 [×4], D30 [×4], D30 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×C4○D12, Dic30 [×4], C4×D15 [×8], D60 [×4], C2×Dic15 [×2], C15⋊7D4 [×8], C2×C60 [×2], C2×C60 [×4], C22×D15 [×2], C22×C30, C2×C4○D20, C2×Dic30, C2×C4×D15 [×2], C2×D60, D60⋊11C2 [×8], C2×C15⋊7D4 [×2], C22×C60, C2×D60⋊11C2
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], D15, C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, D30 [×7], C4○D20 [×2], C23×D5, C2×C4○D12, C22×D15 [×7], C2×C4○D20, D60⋊11C2 [×2], C23×D15, C2×D60⋊11C2
Generators and relations
G = < a,b,c,d | a2=b60=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b30c >
►On 240 points
Generators in S240
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 61)(60 62)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(133 201)(134 202)(135 203)(136 204)(137 205)(138 206)(139 207)(140 208)(141 209)(142 210)(143 211)(144 212)(145 213)(146 214)(147 215)(148 216)(149 217)(150 218)(151 219)(152 220)(153 221)(154 222)(155 223)(156 224)(157 225)(158 226)(159 227)(160 228)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)(169 237)(170 238)(171 239)(172 240)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 64)(62 63)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 114)(72 113)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(121 168)(122 167)(123 166)(124 165)(125 164)(126 163)(127 162)(128 161)(129 160)(130 159)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 184)(182 183)(185 240)(186 239)(187 238)(188 237)(189 236)(190 235)(191 234)(192 233)(193 232)(194 231)(195 230)(196 229)(197 228)(198 227)(199 226)(200 225)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 211)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 218)(22 219)(23 220)(24 221)(25 222)(26 223)(27 224)(28 225)(29 226)(30 227)(31 228)(32 229)(33 230)(34 231)(35 232)(36 233)(37 234)(38 235)(39 236)(40 237)(41 238)(42 239)(43 240)(44 181)(45 182)(46 183)(47 184)(48 185)(49 186)(50 187)(51 188)(52 189)(53 190)(54 191)(55 192)(56 193)(57 194)(58 195)(59 196)(60 197)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 161)(95 162)(96 163)(97 164)(98 165)(99 166)(100 167)(101 168)(102 169)(103 170)(104 171)(105 172)(106 173)(107 174)(108 175)(109 176)(110 177)(111 178)(112 179)(113 180)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
G:=sub<Sym(240)| (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,61)(60,62)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208)(141,209)(142,210)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,221)(154,222)(155,223)(156,224)(157,225)(158,226)(159,227)(160,228)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,237)(170,238)(171,239)(172,240)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,64)(62,63)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,184)(182,183)(185,240)(186,239)(187,238)(188,237)(189,236)(190,235)(191,234)(192,233)(193,232)(194,231)(195,230)(196,229)(197,228)(198,227)(199,226)(200,225)(201,224)(202,223)(203,222)(204,221)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,181)(45,182)(46,183)(47,184)(48,185)(49,186)(50,187)(51,188)(52,189)(53,190)(54,191)(55,192)(56,193)(57,194)(58,195)(59,196)(60,197)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163)(97,164)(98,165)(99,166)(100,167)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127)>;
G:=Group( (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,61)(60,62)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208)(141,209)(142,210)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,221)(154,222)(155,223)(156,224)(157,225)(158,226)(159,227)(160,228)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,237)(170,238)(171,239)(172,240)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,64)(62,63)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,184)(182,183)(185,240)(186,239)(187,238)(188,237)(189,236)(190,235)(191,234)(192,233)(193,232)(194,231)(195,230)(196,229)(197,228)(198,227)(199,226)(200,225)(201,224)(202,223)(203,222)(204,221)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,181)(45,182)(46,183)(47,184)(48,185)(49,186)(50,187)(51,188)(52,189)(53,190)(54,191)(55,192)(56,193)(57,194)(58,195)(59,196)(60,197)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163)(97,164)(98,165)(99,166)(100,167)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127) );
G=PermutationGroup([(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,61),(60,62),(121,189),(122,190),(123,191),(124,192),(125,193),(126,194),(127,195),(128,196),(129,197),(130,198),(131,199),(132,200),(133,201),(134,202),(135,203),(136,204),(137,205),(138,206),(139,207),(140,208),(141,209),(142,210),(143,211),(144,212),(145,213),(146,214),(147,215),(148,216),(149,217),(150,218),(151,219),(152,220),(153,221),(154,222),(155,223),(156,224),(157,225),(158,226),(159,227),(160,228),(161,229),(162,230),(163,231),(164,232),(165,233),(166,234),(167,235),(168,236),(169,237),(170,238),(171,239),(172,240),(173,181),(174,182),(175,183),(176,184),(177,185),(178,186),(179,187),(180,188)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,64),(62,63),(65,120),(66,119),(67,118),(68,117),(69,116),(70,115),(71,114),(72,113),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(121,168),(122,167),(123,166),(124,165),(125,164),(126,163),(127,162),(128,161),(129,160),(130,159),(131,158),(132,157),(133,156),(134,155),(135,154),(136,153),(137,152),(138,151),(139,150),(140,149),(141,148),(142,147),(143,146),(144,145),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,184),(182,183),(185,240),(186,239),(187,238),(188,237),(189,236),(190,235),(191,234),(192,233),(193,232),(194,231),(195,230),(196,229),(197,228),(198,227),(199,226),(200,225),(201,224),(202,223),(203,222),(204,221),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215),(211,214),(212,213)], [(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,211),(15,212),(16,213),(17,214),(18,215),(19,216),(20,217),(21,218),(22,219),(23,220),(24,221),(25,222),(26,223),(27,224),(28,225),(29,226),(30,227),(31,228),(32,229),(33,230),(34,231),(35,232),(36,233),(37,234),(38,235),(39,236),(40,237),(41,238),(42,239),(43,240),(44,181),(45,182),(46,183),(47,184),(48,185),(49,186),(50,187),(51,188),(52,189),(53,190),(54,191),(55,192),(56,193),(57,194),(58,195),(59,196),(60,197),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147),(81,148),(82,149),(83,150),(84,151),(85,152),(86,153),(87,154),(88,155),(89,156),(90,157),(91,158),(92,159),(93,160),(94,161),(95,162),(96,163),(97,164),(98,165),(99,166),(100,167),(101,168),(102,169),(103,170),(104,171),(105,172),(106,173),(107,174),(108,175),(109,176),(110,177),(111,178),(112,179),(113,180),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(120,127)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 37 | 47 | 0 | 0 |
| 14 | 31 | 0 | 0 |
| 0 | 0 | 53 | 28 |
| 0 | 0 | 37 | 0 |
| 25 | 28 | 0 | 0 |
| 30 | 36 | 0 | 0 |
| 0 | 0 | 8 | 33 |
| 0 | 0 | 48 | 53 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 34 | 6 |
| 0 | 0 | 21 | 27 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[37,14,0,0,47,31,0,0,0,0,53,37,0,0,28,0],[25,30,0,0,28,36,0,0,0,0,8,48,0,0,33,53],[60,0,0,0,0,60,0,0,0,0,34,21,0,0,6,27] >;
132 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 | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| 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 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | C4○D12 | D30 | D30 | C4○D20 | D60⋊11C2 |
| kernel | C2×D60⋊11C2 | C2×Dic30 | C2×C4×D15 | C2×D60 | D60⋊11C2 | C2×C15⋊7D4 | C22×C60 | C22×C20 | C22×C12 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 6 | 1 | 4 | 12 | 2 | 4 | 8 | 24 | 4 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_2\times D_{60}\rtimes_{11}C_2 % in TeX
G:=Group("C2xD60:11C2"); // GroupNames label
G:=SmallGroup(480,1168);
// by ID
G=gap.SmallGroup(480,1168);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,2693,18822]);
// 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,b*d=d*b,d*c*d=b^30*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀