metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.8C24, D12.38D10, D30.2C23, C15⋊32- 1+4, C60.161C23, Dic10.41D6, Dic6.40D10, D60.47C22, Dic15.6C23, Dic30.50C22, C4○D12⋊6D5, D15⋊Q8⋊12C2, C3⋊D4.3D10, C5⋊D12.C22, C6.8(C23×D5), C15⋊Q8.3C22, (C6×Dic10)⋊3C2, (C4×S3).14D10, (C2×C20).166D6, C10.8(S3×C23), D12⋊D5⋊11C2, D60⋊11C2⋊6C2, D60⋊C2⋊11C2, D6.3(C22×D5), (S3×Dic10)⋊12C2, (C2×Dic10)⋊13S3, Dic3.D10⋊1C2, (C2×C12).165D10, (S3×C10).3C23, (C2×C60).35C22, (C2×Dic5).69D6, C5⋊1(Q8.15D6), C15⋊7D4.4C22, C3⋊1(D4.10D10), (S3×C20).29C22, C20.126(C22×S3), (C2×C30).227C23, (C4×D15).35C22, (C5×D12).44C22, C12.126(C22×D5), D30.C2.2C22, Dic5.5(C22×S3), Dic3.5(C22×D5), (C5×Dic3).5C23, (C3×Dic5).5C23, (S3×Dic5).2C22, (C5×Dic6).47C22, (C3×Dic10).42C22, (C6×Dic5).128C22, C4.133(C2×S3×D5), (C5×C4○D12)⋊2C2, C22.8(C2×S3×D5), (C2×C4).66(S3×D5), C2.12(C22×S3×D5), (C2×C10).9(C22×S3), (C5×C3⋊D4).3C22, (C2×C6).237(C22×D5), SmallGroup(480,1080)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.C24
G = < a,b,c,d,e | a30=c2=d2=1, b2=e2=a15, bab-1=a19, cac=a11, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a15b, dcd=a15c, ce=ec, de=ed >
Subgroups: 1340 in 292 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×Q8, C5×S3, D15, C30, C30, 2- 1+4, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C4○D12, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, D30, C2×C30, C2×Dic10, C2×Dic10, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, Q8.15D6, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C3×Dic10, C6×Dic5, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D4.10D10, S3×Dic10, D12⋊D5, D60⋊C2, D15⋊Q8, Dic3.D10, C6×Dic10, C5×C4○D12, D60⋊11C2, C30.C24
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2- 1+4, C22×D5, S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5, D4.10D10, C22×S3×D5, C30.C24
►On 240 points
Generators in S240
(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)
(1 131 16 146)(2 150 17 135)(3 139 18 124)(4 128 19 143)(5 147 20 132)(6 136 21 121)(7 125 22 140)(8 144 23 129)(9 133 24 148)(10 122 25 137)(11 141 26 126)(12 130 27 145)(13 149 28 134)(14 138 29 123)(15 127 30 142)(31 161 46 176)(32 180 47 165)(33 169 48 154)(34 158 49 173)(35 177 50 162)(36 166 51 151)(37 155 52 170)(38 174 53 159)(39 163 54 178)(40 152 55 167)(41 171 56 156)(42 160 57 175)(43 179 58 164)(44 168 59 153)(45 157 60 172)(61 200 76 185)(62 189 77 204)(63 208 78 193)(64 197 79 182)(65 186 80 201)(66 205 81 190)(67 194 82 209)(68 183 83 198)(69 202 84 187)(70 191 85 206)(71 210 86 195)(72 199 87 184)(73 188 88 203)(74 207 89 192)(75 196 90 181)(91 237 106 222)(92 226 107 211)(93 215 108 230)(94 234 109 219)(95 223 110 238)(96 212 111 227)(97 231 112 216)(98 220 113 235)(99 239 114 224)(100 228 115 213)(101 217 116 232)(102 236 117 221)(103 225 118 240)(104 214 119 229)(105 233 120 218)
(1 90)(2 71)(3 82)(4 63)(5 74)(6 85)(7 66)(8 77)(9 88)(10 69)(11 80)(12 61)(13 72)(14 83)(15 64)(16 75)(17 86)(18 67)(19 78)(20 89)(21 70)(22 81)(23 62)(24 73)(25 84)(26 65)(27 76)(28 87)(29 68)(30 79)(31 102)(32 113)(33 94)(34 105)(35 116)(36 97)(37 108)(38 119)(39 100)(40 111)(41 92)(42 103)(43 114)(44 95)(45 106)(46 117)(47 98)(48 109)(49 120)(50 101)(51 112)(52 93)(53 104)(54 115)(55 96)(56 107)(57 118)(58 99)(59 110)(60 91)(121 191)(122 202)(123 183)(124 194)(125 205)(126 186)(127 197)(128 208)(129 189)(130 200)(131 181)(132 192)(133 203)(134 184)(135 195)(136 206)(137 187)(138 198)(139 209)(140 190)(141 201)(142 182)(143 193)(144 204)(145 185)(146 196)(147 207)(148 188)(149 199)(150 210)(151 216)(152 227)(153 238)(154 219)(155 230)(156 211)(157 222)(158 233)(159 214)(160 225)(161 236)(162 217)(163 228)(164 239)(165 220)(166 231)(167 212)(168 223)(169 234)(170 215)(171 226)(172 237)(173 218)(174 229)(175 240)(176 221)(177 232)(178 213)(179 224)(180 235)
(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)
(1 36 16 51)(2 37 17 52)(3 38 18 53)(4 39 19 54)(5 40 20 55)(6 41 21 56)(7 42 22 57)(8 43 23 58)(9 44 24 59)(10 45 25 60)(11 46 26 31)(12 47 27 32)(13 48 28 33)(14 49 29 34)(15 50 30 35)(61 98 76 113)(62 99 77 114)(63 100 78 115)(64 101 79 116)(65 102 80 117)(66 103 81 118)(67 104 82 119)(68 105 83 120)(69 106 84 91)(70 107 85 92)(71 108 86 93)(72 109 87 94)(73 110 88 95)(74 111 89 96)(75 112 90 97)(121 171 136 156)(122 172 137 157)(123 173 138 158)(124 174 139 159)(125 175 140 160)(126 176 141 161)(127 177 142 162)(128 178 143 163)(129 179 144 164)(130 180 145 165)(131 151 146 166)(132 152 147 167)(133 153 148 168)(134 154 149 169)(135 155 150 170)(181 216 196 231)(182 217 197 232)(183 218 198 233)(184 219 199 234)(185 220 200 235)(186 221 201 236)(187 222 202 237)(188 223 203 238)(189 224 204 239)(190 225 205 240)(191 226 206 211)(192 227 207 212)(193 228 208 213)(194 229 209 214)(195 230 210 215)
G:=sub<Sym(240)| (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), (1,131,16,146)(2,150,17,135)(3,139,18,124)(4,128,19,143)(5,147,20,132)(6,136,21,121)(7,125,22,140)(8,144,23,129)(9,133,24,148)(10,122,25,137)(11,141,26,126)(12,130,27,145)(13,149,28,134)(14,138,29,123)(15,127,30,142)(31,161,46,176)(32,180,47,165)(33,169,48,154)(34,158,49,173)(35,177,50,162)(36,166,51,151)(37,155,52,170)(38,174,53,159)(39,163,54,178)(40,152,55,167)(41,171,56,156)(42,160,57,175)(43,179,58,164)(44,168,59,153)(45,157,60,172)(61,200,76,185)(62,189,77,204)(63,208,78,193)(64,197,79,182)(65,186,80,201)(66,205,81,190)(67,194,82,209)(68,183,83,198)(69,202,84,187)(70,191,85,206)(71,210,86,195)(72,199,87,184)(73,188,88,203)(74,207,89,192)(75,196,90,181)(91,237,106,222)(92,226,107,211)(93,215,108,230)(94,234,109,219)(95,223,110,238)(96,212,111,227)(97,231,112,216)(98,220,113,235)(99,239,114,224)(100,228,115,213)(101,217,116,232)(102,236,117,221)(103,225,118,240)(104,214,119,229)(105,233,120,218), (1,90)(2,71)(3,82)(4,63)(5,74)(6,85)(7,66)(8,77)(9,88)(10,69)(11,80)(12,61)(13,72)(14,83)(15,64)(16,75)(17,86)(18,67)(19,78)(20,89)(21,70)(22,81)(23,62)(24,73)(25,84)(26,65)(27,76)(28,87)(29,68)(30,79)(31,102)(32,113)(33,94)(34,105)(35,116)(36,97)(37,108)(38,119)(39,100)(40,111)(41,92)(42,103)(43,114)(44,95)(45,106)(46,117)(47,98)(48,109)(49,120)(50,101)(51,112)(52,93)(53,104)(54,115)(55,96)(56,107)(57,118)(58,99)(59,110)(60,91)(121,191)(122,202)(123,183)(124,194)(125,205)(126,186)(127,197)(128,208)(129,189)(130,200)(131,181)(132,192)(133,203)(134,184)(135,195)(136,206)(137,187)(138,198)(139,209)(140,190)(141,201)(142,182)(143,193)(144,204)(145,185)(146,196)(147,207)(148,188)(149,199)(150,210)(151,216)(152,227)(153,238)(154,219)(155,230)(156,211)(157,222)(158,233)(159,214)(160,225)(161,236)(162,217)(163,228)(164,239)(165,220)(166,231)(167,212)(168,223)(169,234)(170,215)(171,226)(172,237)(173,218)(174,229)(175,240)(176,221)(177,232)(178,213)(179,224)(180,235), (61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(181,196)(182,197)(183,198)(184,199)(185,200)(186,201)(187,202)(188,203)(189,204)(190,205)(191,206)(192,207)(193,208)(194,209)(195,210)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240), (1,36,16,51)(2,37,17,52)(3,38,18,53)(4,39,19,54)(5,40,20,55)(6,41,21,56)(7,42,22,57)(8,43,23,58)(9,44,24,59)(10,45,25,60)(11,46,26,31)(12,47,27,32)(13,48,28,33)(14,49,29,34)(15,50,30,35)(61,98,76,113)(62,99,77,114)(63,100,78,115)(64,101,79,116)(65,102,80,117)(66,103,81,118)(67,104,82,119)(68,105,83,120)(69,106,84,91)(70,107,85,92)(71,108,86,93)(72,109,87,94)(73,110,88,95)(74,111,89,96)(75,112,90,97)(121,171,136,156)(122,172,137,157)(123,173,138,158)(124,174,139,159)(125,175,140,160)(126,176,141,161)(127,177,142,162)(128,178,143,163)(129,179,144,164)(130,180,145,165)(131,151,146,166)(132,152,147,167)(133,153,148,168)(134,154,149,169)(135,155,150,170)(181,216,196,231)(182,217,197,232)(183,218,198,233)(184,219,199,234)(185,220,200,235)(186,221,201,236)(187,222,202,237)(188,223,203,238)(189,224,204,239)(190,225,205,240)(191,226,206,211)(192,227,207,212)(193,228,208,213)(194,229,209,214)(195,230,210,215)>;
G:=Group( (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), (1,131,16,146)(2,150,17,135)(3,139,18,124)(4,128,19,143)(5,147,20,132)(6,136,21,121)(7,125,22,140)(8,144,23,129)(9,133,24,148)(10,122,25,137)(11,141,26,126)(12,130,27,145)(13,149,28,134)(14,138,29,123)(15,127,30,142)(31,161,46,176)(32,180,47,165)(33,169,48,154)(34,158,49,173)(35,177,50,162)(36,166,51,151)(37,155,52,170)(38,174,53,159)(39,163,54,178)(40,152,55,167)(41,171,56,156)(42,160,57,175)(43,179,58,164)(44,168,59,153)(45,157,60,172)(61,200,76,185)(62,189,77,204)(63,208,78,193)(64,197,79,182)(65,186,80,201)(66,205,81,190)(67,194,82,209)(68,183,83,198)(69,202,84,187)(70,191,85,206)(71,210,86,195)(72,199,87,184)(73,188,88,203)(74,207,89,192)(75,196,90,181)(91,237,106,222)(92,226,107,211)(93,215,108,230)(94,234,109,219)(95,223,110,238)(96,212,111,227)(97,231,112,216)(98,220,113,235)(99,239,114,224)(100,228,115,213)(101,217,116,232)(102,236,117,221)(103,225,118,240)(104,214,119,229)(105,233,120,218), (1,90)(2,71)(3,82)(4,63)(5,74)(6,85)(7,66)(8,77)(9,88)(10,69)(11,80)(12,61)(13,72)(14,83)(15,64)(16,75)(17,86)(18,67)(19,78)(20,89)(21,70)(22,81)(23,62)(24,73)(25,84)(26,65)(27,76)(28,87)(29,68)(30,79)(31,102)(32,113)(33,94)(34,105)(35,116)(36,97)(37,108)(38,119)(39,100)(40,111)(41,92)(42,103)(43,114)(44,95)(45,106)(46,117)(47,98)(48,109)(49,120)(50,101)(51,112)(52,93)(53,104)(54,115)(55,96)(56,107)(57,118)(58,99)(59,110)(60,91)(121,191)(122,202)(123,183)(124,194)(125,205)(126,186)(127,197)(128,208)(129,189)(130,200)(131,181)(132,192)(133,203)(134,184)(135,195)(136,206)(137,187)(138,198)(139,209)(140,190)(141,201)(142,182)(143,193)(144,204)(145,185)(146,196)(147,207)(148,188)(149,199)(150,210)(151,216)(152,227)(153,238)(154,219)(155,230)(156,211)(157,222)(158,233)(159,214)(160,225)(161,236)(162,217)(163,228)(164,239)(165,220)(166,231)(167,212)(168,223)(169,234)(170,215)(171,226)(172,237)(173,218)(174,229)(175,240)(176,221)(177,232)(178,213)(179,224)(180,235), (61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(181,196)(182,197)(183,198)(184,199)(185,200)(186,201)(187,202)(188,203)(189,204)(190,205)(191,206)(192,207)(193,208)(194,209)(195,210)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240), (1,36,16,51)(2,37,17,52)(3,38,18,53)(4,39,19,54)(5,40,20,55)(6,41,21,56)(7,42,22,57)(8,43,23,58)(9,44,24,59)(10,45,25,60)(11,46,26,31)(12,47,27,32)(13,48,28,33)(14,49,29,34)(15,50,30,35)(61,98,76,113)(62,99,77,114)(63,100,78,115)(64,101,79,116)(65,102,80,117)(66,103,81,118)(67,104,82,119)(68,105,83,120)(69,106,84,91)(70,107,85,92)(71,108,86,93)(72,109,87,94)(73,110,88,95)(74,111,89,96)(75,112,90,97)(121,171,136,156)(122,172,137,157)(123,173,138,158)(124,174,139,159)(125,175,140,160)(126,176,141,161)(127,177,142,162)(128,178,143,163)(129,179,144,164)(130,180,145,165)(131,151,146,166)(132,152,147,167)(133,153,148,168)(134,154,149,169)(135,155,150,170)(181,216,196,231)(182,217,197,232)(183,218,198,233)(184,219,199,234)(185,220,200,235)(186,221,201,236)(187,222,202,237)(188,223,203,238)(189,224,204,239)(190,225,205,240)(191,226,206,211)(192,227,207,212)(193,228,208,213)(194,229,209,214)(195,230,210,215) );
G=PermutationGroup([[(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)], [(1,131,16,146),(2,150,17,135),(3,139,18,124),(4,128,19,143),(5,147,20,132),(6,136,21,121),(7,125,22,140),(8,144,23,129),(9,133,24,148),(10,122,25,137),(11,141,26,126),(12,130,27,145),(13,149,28,134),(14,138,29,123),(15,127,30,142),(31,161,46,176),(32,180,47,165),(33,169,48,154),(34,158,49,173),(35,177,50,162),(36,166,51,151),(37,155,52,170),(38,174,53,159),(39,163,54,178),(40,152,55,167),(41,171,56,156),(42,160,57,175),(43,179,58,164),(44,168,59,153),(45,157,60,172),(61,200,76,185),(62,189,77,204),(63,208,78,193),(64,197,79,182),(65,186,80,201),(66,205,81,190),(67,194,82,209),(68,183,83,198),(69,202,84,187),(70,191,85,206),(71,210,86,195),(72,199,87,184),(73,188,88,203),(74,207,89,192),(75,196,90,181),(91,237,106,222),(92,226,107,211),(93,215,108,230),(94,234,109,219),(95,223,110,238),(96,212,111,227),(97,231,112,216),(98,220,113,235),(99,239,114,224),(100,228,115,213),(101,217,116,232),(102,236,117,221),(103,225,118,240),(104,214,119,229),(105,233,120,218)], [(1,90),(2,71),(3,82),(4,63),(5,74),(6,85),(7,66),(8,77),(9,88),(10,69),(11,80),(12,61),(13,72),(14,83),(15,64),(16,75),(17,86),(18,67),(19,78),(20,89),(21,70),(22,81),(23,62),(24,73),(25,84),(26,65),(27,76),(28,87),(29,68),(30,79),(31,102),(32,113),(33,94),(34,105),(35,116),(36,97),(37,108),(38,119),(39,100),(40,111),(41,92),(42,103),(43,114),(44,95),(45,106),(46,117),(47,98),(48,109),(49,120),(50,101),(51,112),(52,93),(53,104),(54,115),(55,96),(56,107),(57,118),(58,99),(59,110),(60,91),(121,191),(122,202),(123,183),(124,194),(125,205),(126,186),(127,197),(128,208),(129,189),(130,200),(131,181),(132,192),(133,203),(134,184),(135,195),(136,206),(137,187),(138,198),(139,209),(140,190),(141,201),(142,182),(143,193),(144,204),(145,185),(146,196),(147,207),(148,188),(149,199),(150,210),(151,216),(152,227),(153,238),(154,219),(155,230),(156,211),(157,222),(158,233),(159,214),(160,225),(161,236),(162,217),(163,228),(164,239),(165,220),(166,231),(167,212),(168,223),(169,234),(170,215),(171,226),(172,237),(173,218),(174,229),(175,240),(176,221),(177,232),(178,213),(179,224),(180,235)], [(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(181,196),(182,197),(183,198),(184,199),(185,200),(186,201),(187,202),(188,203),(189,204),(190,205),(191,206),(192,207),(193,208),(194,209),(195,210),(211,226),(212,227),(213,228),(214,229),(215,230),(216,231),(217,232),(218,233),(219,234),(220,235),(221,236),(222,237),(223,238),(224,239),(225,240)], [(1,36,16,51),(2,37,17,52),(3,38,18,53),(4,39,19,54),(5,40,20,55),(6,41,21,56),(7,42,22,57),(8,43,23,58),(9,44,24,59),(10,45,25,60),(11,46,26,31),(12,47,27,32),(13,48,28,33),(14,49,29,34),(15,50,30,35),(61,98,76,113),(62,99,77,114),(63,100,78,115),(64,101,79,116),(65,102,80,117),(66,103,81,118),(67,104,82,119),(68,105,83,120),(69,106,84,91),(70,107,85,92),(71,108,86,93),(72,109,87,94),(73,110,88,95),(74,111,89,96),(75,112,90,97),(121,171,136,156),(122,172,137,157),(123,173,138,158),(124,174,139,159),(125,175,140,160),(126,176,141,161),(127,177,142,162),(128,178,143,163),(129,179,144,164),(130,180,145,165),(131,151,146,166),(132,152,147,167),(133,153,148,168),(134,154,149,169),(135,155,150,170),(181,216,196,231),(182,217,197,232),(183,218,198,233),(184,219,199,234),(185,220,200,235),(186,221,201,236),(187,222,202,237),(188,223,203,238),(189,224,204,239),(190,225,205,240),(191,226,206,211),(192,227,207,212),(193,228,208,213),(194,229,209,214),(195,230,210,215)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2- 1+4 | S3×D5 | Q8.15D6 | C2×S3×D5 | C2×S3×D5 | D4.10D10 | C30.C24 |
| kernel | C30.C24 | S3×Dic10 | D12⋊D5 | D60⋊C2 | D15⋊Q8 | Dic3.D10 | C6×Dic10 | C5×C4○D12 | D60⋊11C2 | C2×Dic10 | C4○D12 | Dic10 | C2×Dic5 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of C30.C24 ►in GL4(𝔽61) generated by
| 0 | 14 | 0 | 0 |
| 47 | 53 | 0 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 0 | 13 | 51 |
| 11 | 0 | 0 | 0 |
| 46 | 50 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 46 | 50 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 25 | 57 | 0 | 0 |
| 4 | 36 | 0 | 0 |
| 0 | 0 | 25 | 57 |
| 0 | 0 | 4 | 36 |
G:=sub<GL(4,GF(61))| [0,47,0,0,14,53,0,0,0,0,0,13,0,0,48,51],[11,46,0,0,0,50,0,0,0,0,11,46,0,0,0,50],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[25,4,0,0,57,36,0,0,0,0,25,4,0,0,57,36] >;
C30.C24 in GAP, Magma, Sage, TeX
C_{30}.C_2^4 % in TeX
G:=Group("C30.C2^4"); // GroupNames label
G:=SmallGroup(480,1080);
// by ID
G=gap.SmallGroup(480,1080);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,219,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^30=c^2=d^2=1,b^2=e^2=a^15,b*a*b^-1=a^19,c*a*c=a^11,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^15*b,d*c*d=a^15*c,c*e=e*c,d*e=e*d>;
// generators/relations