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), D60⋊C2⋊11C2, D60⋊11C2⋊6C2, D12⋊D5⋊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, (C2×C30).227C23, C20.126(C22×S3), (C5×D12).44C22, (C4×D15).35C22, C12.126(C22×D5), D30.C2.2C22, (S3×Dic5).2C22, (C3×Dic5).5C23, (C5×Dic3).5C23, Dic5.5(C22×S3), Dic3.5(C22×D5), (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
Subgroups: 1340 in 292 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], C5, S3 [×4], C6, C6, C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], Dic6, Dic6 [×5], C4×S3 [×2], C4×S3 [×10], D12, D12 [×5], C3⋊D4 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C5×S3 [×2], D15 [×2], C30, C30, 2- (1+4), Dic10 [×4], Dic10 [×5], C4×D5 [×6], D20, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C4○D12, C4○D12 [×5], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], C60 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×Dic10, C2×Dic10 [×2], C4○D20 [×3], D4⋊2D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8.15D6, S3×Dic5 [×4], D30.C2 [×4], C5⋊D12 [×4], C15⋊Q8 [×4], C3×Dic10 [×4], C6×Dic5 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, D4.10D10, S3×Dic10 [×2], D12⋊D5 [×2], D60⋊C2 [×2], D15⋊Q8 [×2], Dic3.D10 [×4], C6×Dic10, C5×C4○D12, D60⋊11C2, C30.C24
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, C30.C24
Generators and relations
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 >
►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)
(1 121 16 136)(2 140 17 125)(3 129 18 144)(4 148 19 133)(5 137 20 122)(6 126 21 141)(7 145 22 130)(8 134 23 149)(9 123 24 138)(10 142 25 127)(11 131 26 146)(12 150 27 135)(13 139 28 124)(14 128 29 143)(15 147 30 132)(31 168 46 153)(32 157 47 172)(33 176 48 161)(34 165 49 180)(35 154 50 169)(36 173 51 158)(37 162 52 177)(38 151 53 166)(39 170 54 155)(40 159 55 174)(41 178 56 163)(42 167 57 152)(43 156 58 171)(44 175 59 160)(45 164 60 179)(61 181 76 196)(62 200 77 185)(63 189 78 204)(64 208 79 193)(65 197 80 182)(66 186 81 201)(67 205 82 190)(68 194 83 209)(69 183 84 198)(70 202 85 187)(71 191 86 206)(72 210 87 195)(73 199 88 184)(74 188 89 203)(75 207 90 192)(91 236 106 221)(92 225 107 240)(93 214 108 229)(94 233 109 218)(95 222 110 237)(96 211 111 226)(97 230 112 215)(98 219 113 234)(99 238 114 223)(100 227 115 212)(101 216 116 231)(102 235 117 220)(103 224 118 239)(104 213 119 228)(105 232 120 217)
(1 76)(2 87)(3 68)(4 79)(5 90)(6 71)(7 82)(8 63)(9 74)(10 85)(11 66)(12 77)(13 88)(14 69)(15 80)(16 61)(17 72)(18 83)(19 64)(20 75)(21 86)(22 67)(23 78)(24 89)(25 70)(26 81)(27 62)(28 73)(29 84)(30 65)(31 104)(32 115)(33 96)(34 107)(35 118)(36 99)(37 110)(38 91)(39 102)(40 113)(41 94)(42 105)(43 116)(44 97)(45 108)(46 119)(47 100)(48 111)(49 92)(50 103)(51 114)(52 95)(53 106)(54 117)(55 98)(56 109)(57 120)(58 101)(59 112)(60 93)(121 196)(122 207)(123 188)(124 199)(125 210)(126 191)(127 202)(128 183)(129 194)(130 205)(131 186)(132 197)(133 208)(134 189)(135 200)(136 181)(137 192)(138 203)(139 184)(140 195)(141 206)(142 187)(143 198)(144 209)(145 190)(146 201)(147 182)(148 193)(149 204)(150 185)(151 236)(152 217)(153 228)(154 239)(155 220)(156 231)(157 212)(158 223)(159 234)(160 215)(161 226)(162 237)(163 218)(164 229)(165 240)(166 221)(167 232)(168 213)(169 224)(170 235)(171 216)(172 227)(173 238)(174 219)(175 230)(176 211)(177 222)(178 233)(179 214)(180 225)
(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 53 16 38)(2 54 17 39)(3 55 18 40)(4 56 19 41)(5 57 20 42)(6 58 21 43)(7 59 22 44)(8 60 23 45)(9 31 24 46)(10 32 25 47)(11 33 26 48)(12 34 27 49)(13 35 28 50)(14 36 29 51)(15 37 30 52)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)(121 151 136 166)(122 152 137 167)(123 153 138 168)(124 154 139 169)(125 155 140 170)(126 156 141 171)(127 157 142 172)(128 158 143 173)(129 159 144 174)(130 160 145 175)(131 161 146 176)(132 162 147 177)(133 163 148 178)(134 164 149 179)(135 165 150 180)(181 221 196 236)(182 222 197 237)(183 223 198 238)(184 224 199 239)(185 225 200 240)(186 226 201 211)(187 227 202 212)(188 228 203 213)(189 229 204 214)(190 230 205 215)(191 231 206 216)(192 232 207 217)(193 233 208 218)(194 234 209 219)(195 235 210 220)
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), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,168,46,153)(32,157,47,172)(33,176,48,161)(34,165,49,180)(35,154,50,169)(36,173,51,158)(37,162,52,177)(38,151,53,166)(39,170,54,155)(40,159,55,174)(41,178,56,163)(42,167,57,152)(43,156,58,171)(44,175,59,160)(45,164,60,179)(61,181,76,196)(62,200,77,185)(63,189,78,204)(64,208,79,193)(65,197,80,182)(66,186,81,201)(67,205,82,190)(68,194,83,209)(69,183,84,198)(70,202,85,187)(71,191,86,206)(72,210,87,195)(73,199,88,184)(74,188,89,203)(75,207,90,192)(91,236,106,221)(92,225,107,240)(93,214,108,229)(94,233,109,218)(95,222,110,237)(96,211,111,226)(97,230,112,215)(98,219,113,234)(99,238,114,223)(100,227,115,212)(101,216,116,231)(102,235,117,220)(103,224,118,239)(104,213,119,228)(105,232,120,217), (1,76)(2,87)(3,68)(4,79)(5,90)(6,71)(7,82)(8,63)(9,74)(10,85)(11,66)(12,77)(13,88)(14,69)(15,80)(16,61)(17,72)(18,83)(19,64)(20,75)(21,86)(22,67)(23,78)(24,89)(25,70)(26,81)(27,62)(28,73)(29,84)(30,65)(31,104)(32,115)(33,96)(34,107)(35,118)(36,99)(37,110)(38,91)(39,102)(40,113)(41,94)(42,105)(43,116)(44,97)(45,108)(46,119)(47,100)(48,111)(49,92)(50,103)(51,114)(52,95)(53,106)(54,117)(55,98)(56,109)(57,120)(58,101)(59,112)(60,93)(121,196)(122,207)(123,188)(124,199)(125,210)(126,191)(127,202)(128,183)(129,194)(130,205)(131,186)(132,197)(133,208)(134,189)(135,200)(136,181)(137,192)(138,203)(139,184)(140,195)(141,206)(142,187)(143,198)(144,209)(145,190)(146,201)(147,182)(148,193)(149,204)(150,185)(151,236)(152,217)(153,228)(154,239)(155,220)(156,231)(157,212)(158,223)(159,234)(160,215)(161,226)(162,237)(163,218)(164,229)(165,240)(166,221)(167,232)(168,213)(169,224)(170,235)(171,216)(172,227)(173,238)(174,219)(175,230)(176,211)(177,222)(178,233)(179,214)(180,225), (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), (1,53,16,38)(2,54,17,39)(3,55,18,40)(4,56,19,41)(5,57,20,42)(6,58,21,43)(7,59,22,44)(8,60,23,45)(9,31,24,46)(10,32,25,47)(11,33,26,48)(12,34,27,49)(13,35,28,50)(14,36,29,51)(15,37,30,52)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120)(121,151,136,166)(122,152,137,167)(123,153,138,168)(124,154,139,169)(125,155,140,170)(126,156,141,171)(127,157,142,172)(128,158,143,173)(129,159,144,174)(130,160,145,175)(131,161,146,176)(132,162,147,177)(133,163,148,178)(134,164,149,179)(135,165,150,180)(181,221,196,236)(182,222,197,237)(183,223,198,238)(184,224,199,239)(185,225,200,240)(186,226,201,211)(187,227,202,212)(188,228,203,213)(189,229,204,214)(190,230,205,215)(191,231,206,216)(192,232,207,217)(193,233,208,218)(194,234,209,219)(195,235,210,220)>;
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), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,168,46,153)(32,157,47,172)(33,176,48,161)(34,165,49,180)(35,154,50,169)(36,173,51,158)(37,162,52,177)(38,151,53,166)(39,170,54,155)(40,159,55,174)(41,178,56,163)(42,167,57,152)(43,156,58,171)(44,175,59,160)(45,164,60,179)(61,181,76,196)(62,200,77,185)(63,189,78,204)(64,208,79,193)(65,197,80,182)(66,186,81,201)(67,205,82,190)(68,194,83,209)(69,183,84,198)(70,202,85,187)(71,191,86,206)(72,210,87,195)(73,199,88,184)(74,188,89,203)(75,207,90,192)(91,236,106,221)(92,225,107,240)(93,214,108,229)(94,233,109,218)(95,222,110,237)(96,211,111,226)(97,230,112,215)(98,219,113,234)(99,238,114,223)(100,227,115,212)(101,216,116,231)(102,235,117,220)(103,224,118,239)(104,213,119,228)(105,232,120,217), (1,76)(2,87)(3,68)(4,79)(5,90)(6,71)(7,82)(8,63)(9,74)(10,85)(11,66)(12,77)(13,88)(14,69)(15,80)(16,61)(17,72)(18,83)(19,64)(20,75)(21,86)(22,67)(23,78)(24,89)(25,70)(26,81)(27,62)(28,73)(29,84)(30,65)(31,104)(32,115)(33,96)(34,107)(35,118)(36,99)(37,110)(38,91)(39,102)(40,113)(41,94)(42,105)(43,116)(44,97)(45,108)(46,119)(47,100)(48,111)(49,92)(50,103)(51,114)(52,95)(53,106)(54,117)(55,98)(56,109)(57,120)(58,101)(59,112)(60,93)(121,196)(122,207)(123,188)(124,199)(125,210)(126,191)(127,202)(128,183)(129,194)(130,205)(131,186)(132,197)(133,208)(134,189)(135,200)(136,181)(137,192)(138,203)(139,184)(140,195)(141,206)(142,187)(143,198)(144,209)(145,190)(146,201)(147,182)(148,193)(149,204)(150,185)(151,236)(152,217)(153,228)(154,239)(155,220)(156,231)(157,212)(158,223)(159,234)(160,215)(161,226)(162,237)(163,218)(164,229)(165,240)(166,221)(167,232)(168,213)(169,224)(170,235)(171,216)(172,227)(173,238)(174,219)(175,230)(176,211)(177,222)(178,233)(179,214)(180,225), (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), (1,53,16,38)(2,54,17,39)(3,55,18,40)(4,56,19,41)(5,57,20,42)(6,58,21,43)(7,59,22,44)(8,60,23,45)(9,31,24,46)(10,32,25,47)(11,33,26,48)(12,34,27,49)(13,35,28,50)(14,36,29,51)(15,37,30,52)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120)(121,151,136,166)(122,152,137,167)(123,153,138,168)(124,154,139,169)(125,155,140,170)(126,156,141,171)(127,157,142,172)(128,158,143,173)(129,159,144,174)(130,160,145,175)(131,161,146,176)(132,162,147,177)(133,163,148,178)(134,164,149,179)(135,165,150,180)(181,221,196,236)(182,222,197,237)(183,223,198,238)(184,224,199,239)(185,225,200,240)(186,226,201,211)(187,227,202,212)(188,228,203,213)(189,229,204,214)(190,230,205,215)(191,231,206,216)(192,232,207,217)(193,233,208,218)(194,234,209,219)(195,235,210,220) );
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)], [(1,121,16,136),(2,140,17,125),(3,129,18,144),(4,148,19,133),(5,137,20,122),(6,126,21,141),(7,145,22,130),(8,134,23,149),(9,123,24,138),(10,142,25,127),(11,131,26,146),(12,150,27,135),(13,139,28,124),(14,128,29,143),(15,147,30,132),(31,168,46,153),(32,157,47,172),(33,176,48,161),(34,165,49,180),(35,154,50,169),(36,173,51,158),(37,162,52,177),(38,151,53,166),(39,170,54,155),(40,159,55,174),(41,178,56,163),(42,167,57,152),(43,156,58,171),(44,175,59,160),(45,164,60,179),(61,181,76,196),(62,200,77,185),(63,189,78,204),(64,208,79,193),(65,197,80,182),(66,186,81,201),(67,205,82,190),(68,194,83,209),(69,183,84,198),(70,202,85,187),(71,191,86,206),(72,210,87,195),(73,199,88,184),(74,188,89,203),(75,207,90,192),(91,236,106,221),(92,225,107,240),(93,214,108,229),(94,233,109,218),(95,222,110,237),(96,211,111,226),(97,230,112,215),(98,219,113,234),(99,238,114,223),(100,227,115,212),(101,216,116,231),(102,235,117,220),(103,224,118,239),(104,213,119,228),(105,232,120,217)], [(1,76),(2,87),(3,68),(4,79),(5,90),(6,71),(7,82),(8,63),(9,74),(10,85),(11,66),(12,77),(13,88),(14,69),(15,80),(16,61),(17,72),(18,83),(19,64),(20,75),(21,86),(22,67),(23,78),(24,89),(25,70),(26,81),(27,62),(28,73),(29,84),(30,65),(31,104),(32,115),(33,96),(34,107),(35,118),(36,99),(37,110),(38,91),(39,102),(40,113),(41,94),(42,105),(43,116),(44,97),(45,108),(46,119),(47,100),(48,111),(49,92),(50,103),(51,114),(52,95),(53,106),(54,117),(55,98),(56,109),(57,120),(58,101),(59,112),(60,93),(121,196),(122,207),(123,188),(124,199),(125,210),(126,191),(127,202),(128,183),(129,194),(130,205),(131,186),(132,197),(133,208),(134,189),(135,200),(136,181),(137,192),(138,203),(139,184),(140,195),(141,206),(142,187),(143,198),(144,209),(145,190),(146,201),(147,182),(148,193),(149,204),(150,185),(151,236),(152,217),(153,228),(154,239),(155,220),(156,231),(157,212),(158,223),(159,234),(160,215),(161,226),(162,237),(163,218),(164,229),(165,240),(166,221),(167,232),(168,213),(169,224),(170,235),(171,216),(172,227),(173,238),(174,219),(175,230),(176,211),(177,222),(178,233),(179,214),(180,225)], [(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)], [(1,53,16,38),(2,54,17,39),(3,55,18,40),(4,56,19,41),(5,57,20,42),(6,58,21,43),(7,59,22,44),(8,60,23,45),(9,31,24,46),(10,32,25,47),(11,33,26,48),(12,34,27,49),(13,35,28,50),(14,36,29,51),(15,37,30,52),(61,91,76,106),(62,92,77,107),(63,93,78,108),(64,94,79,109),(65,95,80,110),(66,96,81,111),(67,97,82,112),(68,98,83,113),(69,99,84,114),(70,100,85,115),(71,101,86,116),(72,102,87,117),(73,103,88,118),(74,104,89,119),(75,105,90,120),(121,151,136,166),(122,152,137,167),(123,153,138,168),(124,154,139,169),(125,155,140,170),(126,156,141,171),(127,157,142,172),(128,158,143,173),(129,159,144,174),(130,160,145,175),(131,161,146,176),(132,162,147,177),(133,163,148,178),(134,164,149,179),(135,165,150,180),(181,221,196,236),(182,222,197,237),(183,223,198,238),(184,224,199,239),(185,225,200,240),(186,226,201,211),(187,227,202,212),(188,228,203,213),(189,229,204,214),(190,230,205,215),(191,231,206,216),(192,232,207,217),(193,233,208,218),(194,234,209,219),(195,235,210,220)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀