direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12⋊D5, D12⋊22D10, C30.7C24, Dic10⋊20D6, C60.160C23, D30.33C23, Dic15.35C23, (C2×D12)⋊12D5, C30⋊3(C4○D4), (C10×D12)⋊12C2, C6⋊1(D4⋊2D5), C6.7(C23×D5), (C2×C20).165D6, C5⋊D12⋊8C22, C10.7(S3×C23), D6.2(C22×D5), C10⋊2(Q8⋊3S3), (C6×Dic10)⋊12C2, (C2×Dic10)⋊12S3, (C2×C12).164D10, (C4×D15)⋊22C22, (C5×D12)⋊29C22, (S3×C10).2C23, (S3×Dic5)⋊5C22, (C2×C30).226C23, C20.125(C22×S3), (C2×C60).208C22, (C2×Dic5).136D6, (C22×S3).58D10, C12.125(C22×D5), Dic5.4(C22×S3), (C3×Dic5).4C23, (C3×Dic10)⋊26C22, (C6×Dic5).127C22, (C2×Dic15).232C22, (C22×D15).117C22, C15⋊3(C2×C4○D4), (C2×C4×D15)⋊25C2, C3⋊1(C2×D4⋊2D5), C5⋊2(C2×Q8⋊3S3), C4.132(C2×S3×D5), (C2×S3×Dic5)⋊19C2, (C2×C5⋊D12)⋊17C2, C22.96(C2×S3×D5), C2.11(C22×S3×D5), (C2×C4).218(S3×D5), (S3×C2×C10).58C22, (C2×C6).236(C22×D5), (C2×C10).237(C22×S3), SmallGroup(480,1079)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1532 in 328 conjugacy classes, 116 normal (24 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 [×2], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], C12 [×4], D6 [×4], D6 [×8], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×8], C4×S3 [×12], D12 [×4], D12 [×8], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3 [×2], C22×S3, C5×S3 [×4], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5 [×2], C2×Dic5 [×9], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], S3×C2×C4 [×3], C2×D12, C2×D12 [×2], Q8⋊3S3 [×8], C6×Q8, C3×Dic5 [×4], Dic15 [×2], C60 [×2], S3×C10 [×4], S3×C10 [×4], D30 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5, D4⋊2D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×Q8⋊3S3, S3×Dic5 [×8], C5⋊D12 [×8], C3×Dic10 [×4], C6×Dic5 [×2], C5×D12 [×4], C4×D15 [×4], C2×Dic15, C2×C60, S3×C2×C10 [×2], C22×D15, C2×D4⋊2D5, D12⋊D5 [×8], C2×S3×Dic5 [×2], C2×C5⋊D12 [×2], C6×Dic10, C10×D12, C2×C4×D15, C2×D12⋊D5
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, D4⋊2D5 [×2], C23×D5, C2×Q8⋊3S3, C2×S3×D5 [×3], C2×D4⋊2D5, D12⋊D5 [×2], C22×S3×D5, C2×D12⋊D5
Generators and relations
G = < a,b,c,d,e | a2=b12=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ece=b10c, ede=d-1 >
►On 240 points
Generators in S240
(1 153)(2 154)(3 155)(4 156)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 121)(24 122)(25 120)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 94)(38 95)(39 96)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 133)(81 134)(82 135)(83 136)(84 137)(97 210)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 205)(105 206)(106 207)(107 208)(108 209)(157 231)(158 232)(159 233)(160 234)(161 235)(162 236)(163 237)(164 238)(165 239)(166 240)(167 229)(168 230)(169 187)(170 188)(171 189)(172 190)(173 191)(174 192)(175 181)(176 182)(177 183)(178 184)(179 185)(180 186)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)
(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)
(1 152)(2 151)(3 150)(4 149)(5 148)(6 147)(7 146)(8 145)(9 156)(10 155)(11 154)(12 153)(13 130)(14 129)(15 128)(16 127)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 132)(24 131)(25 111)(26 110)(27 109)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 65)(50 64)(51 63)(52 62)(53 61)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 144)(81 143)(82 142)(83 141)(84 140)(97 213)(98 212)(99 211)(100 210)(101 209)(102 208)(103 207)(104 206)(105 205)(106 216)(107 215)(108 214)(157 230)(158 229)(159 240)(160 239)(161 238)(162 237)(163 236)(164 235)(165 234)(166 233)(167 232)(168 231)(169 184)(170 183)(171 182)(172 181)(173 192)(174 191)(175 190)(176 189)(177 188)(178 187)(179 186)(180 185)(193 220)(194 219)(195 218)(196 217)(197 228)(198 227)(199 226)(200 225)(201 224)(202 223)(203 222)(204 221)
(1 212 74 23 59)(2 213 75 24 60)(3 214 76 13 49)(4 215 77 14 50)(5 216 78 15 51)(6 205 79 16 52)(7 206 80 17 53)(8 207 81 18 54)(9 208 82 19 55)(10 209 83 20 56)(11 210 84 21 57)(12 211 73 22 58)(25 161 48 190 220)(26 162 37 191 221)(27 163 38 192 222)(28 164 39 181 223)(29 165 40 182 224)(30 166 41 183 225)(31 167 42 184 226)(32 168 43 185 227)(33 157 44 186 228)(34 158 45 187 217)(35 159 46 188 218)(36 160 47 189 219)(61 146 104 144 126)(62 147 105 133 127)(63 148 106 134 128)(64 149 107 135 129)(65 150 108 136 130)(66 151 97 137 131)(67 152 98 138 132)(68 153 99 139 121)(69 154 100 140 122)(70 155 101 141 123)(71 156 102 142 124)(72 145 103 143 125)(85 176 194 112 239)(86 177 195 113 240)(87 178 196 114 229)(88 179 197 115 230)(89 180 198 116 231)(90 169 199 117 232)(91 170 200 118 233)(92 171 201 119 234)(93 172 202 120 235)(94 173 203 109 236)(95 174 204 110 237)(96 175 193 111 238)
(1 119)(2 112)(3 117)(4 110)(5 115)(6 120)(7 113)(8 118)(9 111)(10 116)(11 109)(12 114)(13 90)(14 95)(15 88)(16 93)(17 86)(18 91)(19 96)(20 89)(21 94)(22 87)(23 92)(24 85)(25 146)(26 151)(27 156)(28 149)(29 154)(30 147)(31 152)(32 145)(33 150)(34 155)(35 148)(36 153)(37 131)(38 124)(39 129)(40 122)(41 127)(42 132)(43 125)(44 130)(45 123)(46 128)(47 121)(48 126)(49 232)(50 237)(51 230)(52 235)(53 240)(54 233)(55 238)(56 231)(57 236)(58 229)(59 234)(60 239)(61 161)(62 166)(63 159)(64 164)(65 157)(66 162)(67 167)(68 160)(69 165)(70 158)(71 163)(72 168)(73 178)(74 171)(75 176)(76 169)(77 174)(78 179)(79 172)(80 177)(81 170)(82 175)(83 180)(84 173)(97 221)(98 226)(99 219)(100 224)(101 217)(102 222)(103 227)(104 220)(105 225)(106 218)(107 223)(108 228)(133 183)(134 188)(135 181)(136 186)(137 191)(138 184)(139 189)(140 182)(141 187)(142 192)(143 185)(144 190)(193 208)(194 213)(195 206)(196 211)(197 216)(198 209)(199 214)(200 207)(201 212)(202 205)(203 210)(204 215)
G:=sub<Sym(240)| (1,153)(2,154)(3,155)(4,156)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,121)(24,122)(25,120)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,133)(81,134)(82,135)(83,136)(84,137)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,205)(105,206)(106,207)(107,208)(108,209)(157,231)(158,232)(159,233)(160,234)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,229)(168,230)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222), (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), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,156)(10,155)(11,154)(12,153)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,132)(24,131)(25,111)(26,110)(27,109)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,144)(81,143)(82,142)(83,141)(84,140)(97,213)(98,212)(99,211)(100,210)(101,209)(102,208)(103,207)(104,206)(105,205)(106,216)(107,215)(108,214)(157,230)(158,229)(159,240)(160,239)(161,238)(162,237)(163,236)(164,235)(165,234)(166,233)(167,232)(168,231)(169,184)(170,183)(171,182)(172,181)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(193,220)(194,219)(195,218)(196,217)(197,228)(198,227)(199,226)(200,225)(201,224)(202,223)(203,222)(204,221), (1,212,74,23,59)(2,213,75,24,60)(3,214,76,13,49)(4,215,77,14,50)(5,216,78,15,51)(6,205,79,16,52)(7,206,80,17,53)(8,207,81,18,54)(9,208,82,19,55)(10,209,83,20,56)(11,210,84,21,57)(12,211,73,22,58)(25,161,48,190,220)(26,162,37,191,221)(27,163,38,192,222)(28,164,39,181,223)(29,165,40,182,224)(30,166,41,183,225)(31,167,42,184,226)(32,168,43,185,227)(33,157,44,186,228)(34,158,45,187,217)(35,159,46,188,218)(36,160,47,189,219)(61,146,104,144,126)(62,147,105,133,127)(63,148,106,134,128)(64,149,107,135,129)(65,150,108,136,130)(66,151,97,137,131)(67,152,98,138,132)(68,153,99,139,121)(69,154,100,140,122)(70,155,101,141,123)(71,156,102,142,124)(72,145,103,143,125)(85,176,194,112,239)(86,177,195,113,240)(87,178,196,114,229)(88,179,197,115,230)(89,180,198,116,231)(90,169,199,117,232)(91,170,200,118,233)(92,171,201,119,234)(93,172,202,120,235)(94,173,203,109,236)(95,174,204,110,237)(96,175,193,111,238), (1,119)(2,112)(3,117)(4,110)(5,115)(6,120)(7,113)(8,118)(9,111)(10,116)(11,109)(12,114)(13,90)(14,95)(15,88)(16,93)(17,86)(18,91)(19,96)(20,89)(21,94)(22,87)(23,92)(24,85)(25,146)(26,151)(27,156)(28,149)(29,154)(30,147)(31,152)(32,145)(33,150)(34,155)(35,148)(36,153)(37,131)(38,124)(39,129)(40,122)(41,127)(42,132)(43,125)(44,130)(45,123)(46,128)(47,121)(48,126)(49,232)(50,237)(51,230)(52,235)(53,240)(54,233)(55,238)(56,231)(57,236)(58,229)(59,234)(60,239)(61,161)(62,166)(63,159)(64,164)(65,157)(66,162)(67,167)(68,160)(69,165)(70,158)(71,163)(72,168)(73,178)(74,171)(75,176)(76,169)(77,174)(78,179)(79,172)(80,177)(81,170)(82,175)(83,180)(84,173)(97,221)(98,226)(99,219)(100,224)(101,217)(102,222)(103,227)(104,220)(105,225)(106,218)(107,223)(108,228)(133,183)(134,188)(135,181)(136,186)(137,191)(138,184)(139,189)(140,182)(141,187)(142,192)(143,185)(144,190)(193,208)(194,213)(195,206)(196,211)(197,216)(198,209)(199,214)(200,207)(201,212)(202,205)(203,210)(204,215)>;
G:=Group( (1,153)(2,154)(3,155)(4,156)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,121)(24,122)(25,120)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,133)(81,134)(82,135)(83,136)(84,137)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,205)(105,206)(106,207)(107,208)(108,209)(157,231)(158,232)(159,233)(160,234)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,229)(168,230)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222), (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), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,156)(10,155)(11,154)(12,153)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,132)(24,131)(25,111)(26,110)(27,109)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,144)(81,143)(82,142)(83,141)(84,140)(97,213)(98,212)(99,211)(100,210)(101,209)(102,208)(103,207)(104,206)(105,205)(106,216)(107,215)(108,214)(157,230)(158,229)(159,240)(160,239)(161,238)(162,237)(163,236)(164,235)(165,234)(166,233)(167,232)(168,231)(169,184)(170,183)(171,182)(172,181)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(193,220)(194,219)(195,218)(196,217)(197,228)(198,227)(199,226)(200,225)(201,224)(202,223)(203,222)(204,221), (1,212,74,23,59)(2,213,75,24,60)(3,214,76,13,49)(4,215,77,14,50)(5,216,78,15,51)(6,205,79,16,52)(7,206,80,17,53)(8,207,81,18,54)(9,208,82,19,55)(10,209,83,20,56)(11,210,84,21,57)(12,211,73,22,58)(25,161,48,190,220)(26,162,37,191,221)(27,163,38,192,222)(28,164,39,181,223)(29,165,40,182,224)(30,166,41,183,225)(31,167,42,184,226)(32,168,43,185,227)(33,157,44,186,228)(34,158,45,187,217)(35,159,46,188,218)(36,160,47,189,219)(61,146,104,144,126)(62,147,105,133,127)(63,148,106,134,128)(64,149,107,135,129)(65,150,108,136,130)(66,151,97,137,131)(67,152,98,138,132)(68,153,99,139,121)(69,154,100,140,122)(70,155,101,141,123)(71,156,102,142,124)(72,145,103,143,125)(85,176,194,112,239)(86,177,195,113,240)(87,178,196,114,229)(88,179,197,115,230)(89,180,198,116,231)(90,169,199,117,232)(91,170,200,118,233)(92,171,201,119,234)(93,172,202,120,235)(94,173,203,109,236)(95,174,204,110,237)(96,175,193,111,238), (1,119)(2,112)(3,117)(4,110)(5,115)(6,120)(7,113)(8,118)(9,111)(10,116)(11,109)(12,114)(13,90)(14,95)(15,88)(16,93)(17,86)(18,91)(19,96)(20,89)(21,94)(22,87)(23,92)(24,85)(25,146)(26,151)(27,156)(28,149)(29,154)(30,147)(31,152)(32,145)(33,150)(34,155)(35,148)(36,153)(37,131)(38,124)(39,129)(40,122)(41,127)(42,132)(43,125)(44,130)(45,123)(46,128)(47,121)(48,126)(49,232)(50,237)(51,230)(52,235)(53,240)(54,233)(55,238)(56,231)(57,236)(58,229)(59,234)(60,239)(61,161)(62,166)(63,159)(64,164)(65,157)(66,162)(67,167)(68,160)(69,165)(70,158)(71,163)(72,168)(73,178)(74,171)(75,176)(76,169)(77,174)(78,179)(79,172)(80,177)(81,170)(82,175)(83,180)(84,173)(97,221)(98,226)(99,219)(100,224)(101,217)(102,222)(103,227)(104,220)(105,225)(106,218)(107,223)(108,228)(133,183)(134,188)(135,181)(136,186)(137,191)(138,184)(139,189)(140,182)(141,187)(142,192)(143,185)(144,190)(193,208)(194,213)(195,206)(196,211)(197,216)(198,209)(199,214)(200,207)(201,212)(202,205)(203,210)(204,215) );
G=PermutationGroup([(1,153),(2,154),(3,155),(4,156),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,121),(24,122),(25,120),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,94),(38,95),(39,96),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(79,144),(80,133),(81,134),(82,135),(83,136),(84,137),(97,210),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,205),(105,206),(106,207),(107,208),(108,209),(157,231),(158,232),(159,233),(160,234),(161,235),(162,236),(163,237),(164,238),(165,239),(166,240),(167,229),(168,230),(169,187),(170,188),(171,189),(172,190),(173,191),(174,192),(175,181),(176,182),(177,183),(178,184),(179,185),(180,186),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,217),(200,218),(201,219),(202,220),(203,221),(204,222)], [(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)], [(1,152),(2,151),(3,150),(4,149),(5,148),(6,147),(7,146),(8,145),(9,156),(10,155),(11,154),(12,153),(13,130),(14,129),(15,128),(16,127),(17,126),(18,125),(19,124),(20,123),(21,122),(22,121),(23,132),(24,131),(25,111),(26,110),(27,109),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,96),(49,65),(50,64),(51,63),(52,62),(53,61),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(73,139),(74,138),(75,137),(76,136),(77,135),(78,134),(79,133),(80,144),(81,143),(82,142),(83,141),(84,140),(97,213),(98,212),(99,211),(100,210),(101,209),(102,208),(103,207),(104,206),(105,205),(106,216),(107,215),(108,214),(157,230),(158,229),(159,240),(160,239),(161,238),(162,237),(163,236),(164,235),(165,234),(166,233),(167,232),(168,231),(169,184),(170,183),(171,182),(172,181),(173,192),(174,191),(175,190),(176,189),(177,188),(178,187),(179,186),(180,185),(193,220),(194,219),(195,218),(196,217),(197,228),(198,227),(199,226),(200,225),(201,224),(202,223),(203,222),(204,221)], [(1,212,74,23,59),(2,213,75,24,60),(3,214,76,13,49),(4,215,77,14,50),(5,216,78,15,51),(6,205,79,16,52),(7,206,80,17,53),(8,207,81,18,54),(9,208,82,19,55),(10,209,83,20,56),(11,210,84,21,57),(12,211,73,22,58),(25,161,48,190,220),(26,162,37,191,221),(27,163,38,192,222),(28,164,39,181,223),(29,165,40,182,224),(30,166,41,183,225),(31,167,42,184,226),(32,168,43,185,227),(33,157,44,186,228),(34,158,45,187,217),(35,159,46,188,218),(36,160,47,189,219),(61,146,104,144,126),(62,147,105,133,127),(63,148,106,134,128),(64,149,107,135,129),(65,150,108,136,130),(66,151,97,137,131),(67,152,98,138,132),(68,153,99,139,121),(69,154,100,140,122),(70,155,101,141,123),(71,156,102,142,124),(72,145,103,143,125),(85,176,194,112,239),(86,177,195,113,240),(87,178,196,114,229),(88,179,197,115,230),(89,180,198,116,231),(90,169,199,117,232),(91,170,200,118,233),(92,171,201,119,234),(93,172,202,120,235),(94,173,203,109,236),(95,174,204,110,237),(96,175,193,111,238)], [(1,119),(2,112),(3,117),(4,110),(5,115),(6,120),(7,113),(8,118),(9,111),(10,116),(11,109),(12,114),(13,90),(14,95),(15,88),(16,93),(17,86),(18,91),(19,96),(20,89),(21,94),(22,87),(23,92),(24,85),(25,146),(26,151),(27,156),(28,149),(29,154),(30,147),(31,152),(32,145),(33,150),(34,155),(35,148),(36,153),(37,131),(38,124),(39,129),(40,122),(41,127),(42,132),(43,125),(44,130),(45,123),(46,128),(47,121),(48,126),(49,232),(50,237),(51,230),(52,235),(53,240),(54,233),(55,238),(56,231),(57,236),(58,229),(59,234),(60,239),(61,161),(62,166),(63,159),(64,164),(65,157),(66,162),(67,167),(68,160),(69,165),(70,158),(71,163),(72,168),(73,178),(74,171),(75,176),(76,169),(77,174),(78,179),(79,172),(80,177),(81,170),(82,175),(83,180),(84,173),(97,221),(98,226),(99,219),(100,224),(101,217),(102,222),(103,227),(104,220),(105,225),(106,218),(107,223),(108,228),(133,183),(134,188),(135,181),(136,186),(137,191),(138,184),(139,189),(140,182),(141,187),(142,192),(143,185),(144,190),(193,208),(194,213),(195,206),(196,211),(197,216),(198,209),(199,214),(200,207),(201,212),(202,205),(203,210),(204,215)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 52 |
| 0 | 0 | 0 | 0 | 15 | 45 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 17 |
| 0 | 0 | 0 | 0 | 46 | 16 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 19 | 60 | 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 | 1 |
| 28 | 53 | 0 | 0 | 0 | 0 |
| 14 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 23 |
| 0 | 0 | 0 | 0 | 43 | 7 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,16,15,0,0,0,0,52,45],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,45,46,0,0,0,0,17,16],[18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,14,0,0,0,0,53,33,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,54,43,0,0,0,0,23,7] >;
66 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 | 20B | 20C | 20D | 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 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | C2×S3×D5 | D12⋊D5 |
| kernel | C2×D12⋊D5 | D12⋊D5 | C2×S3×Dic5 | C2×C5⋊D12 | C6×Dic10 | C10×D12 | C2×C4×D15 | C2×Dic10 | C2×D12 | Dic10 | C2×Dic5 | C2×C20 | C30 | D12 | C2×C12 | C22×S3 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 4 | 8 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{12}\rtimes D_5 % in TeX
G:=Group("C2xD12:D5"); // GroupNames label
G:=SmallGroup(480,1079);
// by ID
G=gap.SmallGroup(480,1079);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations
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