direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C42⋊3S3, (C4×C20)⋊4S3, (C4×C60)⋊2C2, (C4×C12)⋊1C10, C42⋊3(C5×S3), D6⋊C4.1C10, Dic3⋊C4⋊1C10, (C2×C20).374D6, C15⋊17(C42⋊2C2), C30.202(C4○D4), (C2×C30).396C23, (C2×C60).454C22, C10.113(C4○D12), (C10×Dic3).137C22, C6.6(C5×C4○D4), (C5×D6⋊C4).1C2, C3⋊1(C5×C42⋊2C2), C2.8(C5×C4○D12), (C2×C4).66(S3×C10), (C5×Dic3⋊C4)⋊1C2, C22.38(S3×C2×C10), (C2×C12).73(C2×C10), (S3×C2×C10).65C22, (C22×S3).3(C2×C10), (C2×C6).17(C22×C10), (C2×Dic3).4(C2×C10), (C2×C10).330(C22×S3), SmallGroup(480,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊3S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c-1, ede=d-1 >
Subgroups: 276 in 120 conjugacy classes, 58 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C42⋊2C2, C2×C20, C2×C20, C22×C10, Dic3⋊C4, D6⋊C4, C4×C12, C5×Dic3, C60, S3×C10, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C42⋊3S3, C10×Dic3, C2×C60, S3×C2×C10, C5×C42⋊2C2, C5×Dic3⋊C4, C5×D6⋊C4, C4×C60, C5×C42⋊3S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C42⋊2C2, C22×C10, C4○D12, S3×C10, C5×C4○D4, C42⋊3S3, S3×C2×C10, C5×C42⋊2C2, C5×C4○D12, C5×C42⋊3S3
►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)
(1 186 66 141)(2 187 67 142)(3 188 68 143)(4 189 69 144)(5 190 70 145)(6 170 52 151)(7 166 53 152)(8 167 54 153)(9 168 55 154)(10 169 51 155)(11 125 231 111)(12 121 232 112)(13 122 233 113)(14 123 234 114)(15 124 235 115)(16 206 86 134)(17 207 87 135)(18 208 88 131)(19 209 89 132)(20 210 90 133)(21 147 195 75)(22 148 191 71)(23 149 192 72)(24 150 193 73)(25 146 194 74)(26 178 58 163)(27 179 59 164)(28 180 60 165)(29 176 56 161)(30 177 57 162)(31 183 63 174)(32 184 64 175)(33 185 65 171)(34 181 61 172)(35 182 62 173)(36 126 201 81)(37 127 202 82)(38 128 203 83)(39 129 204 84)(40 130 205 85)(41 199 79 159)(42 200 80 160)(43 196 76 156)(44 197 77 157)(45 198 78 158)(46 136 216 96)(47 137 217 97)(48 138 218 98)(49 139 219 99)(50 140 220 100)(91 230 110 211)(92 226 106 212)(93 227 107 213)(94 228 108 214)(95 229 109 215)(101 236 116 221)(102 237 117 222)(103 238 118 223)(104 239 119 224)(105 240 120 225)
(1 118 58 81)(2 119 59 82)(3 120 60 83)(4 116 56 84)(5 117 57 85)(6 150 90 91)(7 146 86 92)(8 147 87 93)(9 148 88 94)(10 149 89 95)(11 185 48 198)(12 181 49 199)(13 182 50 200)(14 183 46 196)(15 184 47 197)(16 106 53 74)(17 107 54 75)(18 108 55 71)(19 109 51 72)(20 110 52 73)(21 207 213 153)(22 208 214 154)(23 209 215 155)(24 210 211 151)(25 206 212 152)(26 126 66 103)(27 127 67 104)(28 128 68 105)(29 129 69 101)(30 130 70 102)(31 96 43 114)(32 97 44 115)(33 98 45 111)(34 99 41 112)(35 100 42 113)(36 186 223 163)(37 187 224 164)(38 188 225 165)(39 189 221 161)(40 190 222 162)(61 139 79 121)(62 140 80 122)(63 136 76 123)(64 137 77 124)(65 138 78 125)(131 228 168 191)(132 229 169 192)(133 230 170 193)(134 226 166 194)(135 227 167 195)(141 238 178 201)(142 239 179 202)(143 240 180 203)(144 236 176 204)(145 237 177 205)(156 234 174 216)(157 235 175 217)(158 231 171 218)(159 232 172 219)(160 233 173 220)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 193 36)(12 194 37)(13 195 38)(14 191 39)(15 192 40)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 203 233)(22 204 234)(23 205 235)(24 201 231)(25 202 232)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 228 221)(47 229 222)(48 230 223)(49 226 224)(50 227 225)(71 129 123)(72 130 124)(73 126 125)(74 127 121)(75 128 122)(81 111 150)(82 112 146)(83 113 147)(84 114 148)(85 115 149)(91 118 98)(92 119 99)(93 120 100)(94 116 96)(95 117 97)(101 136 108)(102 137 109)(103 138 110)(104 139 106)(105 140 107)(131 189 183)(132 190 184)(133 186 185)(134 187 181)(135 188 182)(141 171 210)(142 172 206)(143 173 207)(144 174 208)(145 175 209)(151 178 158)(152 179 159)(153 180 160)(154 176 156)(155 177 157)(161 196 168)(162 197 169)(163 198 170)(164 199 166)(165 200 167)(211 238 218)(212 239 219)(213 240 220)(214 236 216)(215 237 217)
(6 45)(7 41)(8 42)(9 43)(10 44)(11 24)(12 25)(13 21)(14 22)(15 23)(16 61)(17 62)(18 63)(19 64)(20 65)(31 88)(32 89)(33 90)(34 86)(35 87)(36 201)(37 202)(38 203)(39 204)(40 205)(46 214)(47 215)(48 211)(49 212)(50 213)(51 77)(52 78)(53 79)(54 80)(55 76)(71 96)(72 97)(73 98)(74 99)(75 100)(81 103)(82 104)(83 105)(84 101)(85 102)(91 125)(92 121)(93 122)(94 123)(95 124)(106 112)(107 113)(108 114)(109 115)(110 111)(116 129)(117 130)(118 126)(119 127)(120 128)(131 196)(132 197)(133 198)(134 199)(135 200)(136 148)(137 149)(138 150)(139 146)(140 147)(141 178)(142 179)(143 180)(144 176)(145 177)(151 171)(152 172)(153 173)(154 174)(155 175)(156 208)(157 209)(158 210)(159 206)(160 207)(161 189)(162 190)(163 186)(164 187)(165 188)(166 181)(167 182)(168 183)(169 184)(170 185)(191 234)(192 235)(193 231)(194 232)(195 233)(216 228)(217 229)(218 230)(219 226)(220 227)(221 236)(222 237)(223 238)(224 239)(225 240)
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), (1,186,66,141)(2,187,67,142)(3,188,68,143)(4,189,69,144)(5,190,70,145)(6,170,52,151)(7,166,53,152)(8,167,54,153)(9,168,55,154)(10,169,51,155)(11,125,231,111)(12,121,232,112)(13,122,233,113)(14,123,234,114)(15,124,235,115)(16,206,86,134)(17,207,87,135)(18,208,88,131)(19,209,89,132)(20,210,90,133)(21,147,195,75)(22,148,191,71)(23,149,192,72)(24,150,193,73)(25,146,194,74)(26,178,58,163)(27,179,59,164)(28,180,60,165)(29,176,56,161)(30,177,57,162)(31,183,63,174)(32,184,64,175)(33,185,65,171)(34,181,61,172)(35,182,62,173)(36,126,201,81)(37,127,202,82)(38,128,203,83)(39,129,204,84)(40,130,205,85)(41,199,79,159)(42,200,80,160)(43,196,76,156)(44,197,77,157)(45,198,78,158)(46,136,216,96)(47,137,217,97)(48,138,218,98)(49,139,219,99)(50,140,220,100)(91,230,110,211)(92,226,106,212)(93,227,107,213)(94,228,108,214)(95,229,109,215)(101,236,116,221)(102,237,117,222)(103,238,118,223)(104,239,119,224)(105,240,120,225), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,150,90,91)(7,146,86,92)(8,147,87,93)(9,148,88,94)(10,149,89,95)(11,185,48,198)(12,181,49,199)(13,182,50,200)(14,183,46,196)(15,184,47,197)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(21,207,213,153)(22,208,214,154)(23,209,215,155)(24,210,211,151)(25,206,212,152)(26,126,66,103)(27,127,67,104)(28,128,68,105)(29,129,69,101)(30,130,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113)(36,186,223,163)(37,187,224,164)(38,188,225,165)(39,189,221,161)(40,190,222,162)(61,139,79,121)(62,140,80,122)(63,136,76,123)(64,137,77,124)(65,138,78,125)(131,228,168,191)(132,229,169,192)(133,230,170,193)(134,226,166,194)(135,227,167,195)(141,238,178,201)(142,239,179,202)(143,240,180,203)(144,236,176,204)(145,237,177,205)(156,234,174,216)(157,235,175,217)(158,231,171,218)(159,232,172,219)(160,233,173,220), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,193,36)(12,194,37)(13,195,38)(14,191,39)(15,192,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,203,233)(22,204,234)(23,205,235)(24,201,231)(25,202,232)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,228,221)(47,229,222)(48,230,223)(49,226,224)(50,227,225)(71,129,123)(72,130,124)(73,126,125)(74,127,121)(75,128,122)(81,111,150)(82,112,146)(83,113,147)(84,114,148)(85,115,149)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97)(101,136,108)(102,137,109)(103,138,110)(104,139,106)(105,140,107)(131,189,183)(132,190,184)(133,186,185)(134,187,181)(135,188,182)(141,171,210)(142,172,206)(143,173,207)(144,174,208)(145,175,209)(151,178,158)(152,179,159)(153,180,160)(154,176,156)(155,177,157)(161,196,168)(162,197,169)(163,198,170)(164,199,166)(165,200,167)(211,238,218)(212,239,219)(213,240,220)(214,236,216)(215,237,217), (6,45)(7,41)(8,42)(9,43)(10,44)(11,24)(12,25)(13,21)(14,22)(15,23)(16,61)(17,62)(18,63)(19,64)(20,65)(31,88)(32,89)(33,90)(34,86)(35,87)(36,201)(37,202)(38,203)(39,204)(40,205)(46,214)(47,215)(48,211)(49,212)(50,213)(51,77)(52,78)(53,79)(54,80)(55,76)(71,96)(72,97)(73,98)(74,99)(75,100)(81,103)(82,104)(83,105)(84,101)(85,102)(91,125)(92,121)(93,122)(94,123)(95,124)(106,112)(107,113)(108,114)(109,115)(110,111)(116,129)(117,130)(118,126)(119,127)(120,128)(131,196)(132,197)(133,198)(134,199)(135,200)(136,148)(137,149)(138,150)(139,146)(140,147)(141,178)(142,179)(143,180)(144,176)(145,177)(151,171)(152,172)(153,173)(154,174)(155,175)(156,208)(157,209)(158,210)(159,206)(160,207)(161,189)(162,190)(163,186)(164,187)(165,188)(166,181)(167,182)(168,183)(169,184)(170,185)(191,234)(192,235)(193,231)(194,232)(195,233)(216,228)(217,229)(218,230)(219,226)(220,227)(221,236)(222,237)(223,238)(224,239)(225,240)>;
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), (1,186,66,141)(2,187,67,142)(3,188,68,143)(4,189,69,144)(5,190,70,145)(6,170,52,151)(7,166,53,152)(8,167,54,153)(9,168,55,154)(10,169,51,155)(11,125,231,111)(12,121,232,112)(13,122,233,113)(14,123,234,114)(15,124,235,115)(16,206,86,134)(17,207,87,135)(18,208,88,131)(19,209,89,132)(20,210,90,133)(21,147,195,75)(22,148,191,71)(23,149,192,72)(24,150,193,73)(25,146,194,74)(26,178,58,163)(27,179,59,164)(28,180,60,165)(29,176,56,161)(30,177,57,162)(31,183,63,174)(32,184,64,175)(33,185,65,171)(34,181,61,172)(35,182,62,173)(36,126,201,81)(37,127,202,82)(38,128,203,83)(39,129,204,84)(40,130,205,85)(41,199,79,159)(42,200,80,160)(43,196,76,156)(44,197,77,157)(45,198,78,158)(46,136,216,96)(47,137,217,97)(48,138,218,98)(49,139,219,99)(50,140,220,100)(91,230,110,211)(92,226,106,212)(93,227,107,213)(94,228,108,214)(95,229,109,215)(101,236,116,221)(102,237,117,222)(103,238,118,223)(104,239,119,224)(105,240,120,225), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,150,90,91)(7,146,86,92)(8,147,87,93)(9,148,88,94)(10,149,89,95)(11,185,48,198)(12,181,49,199)(13,182,50,200)(14,183,46,196)(15,184,47,197)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(21,207,213,153)(22,208,214,154)(23,209,215,155)(24,210,211,151)(25,206,212,152)(26,126,66,103)(27,127,67,104)(28,128,68,105)(29,129,69,101)(30,130,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113)(36,186,223,163)(37,187,224,164)(38,188,225,165)(39,189,221,161)(40,190,222,162)(61,139,79,121)(62,140,80,122)(63,136,76,123)(64,137,77,124)(65,138,78,125)(131,228,168,191)(132,229,169,192)(133,230,170,193)(134,226,166,194)(135,227,167,195)(141,238,178,201)(142,239,179,202)(143,240,180,203)(144,236,176,204)(145,237,177,205)(156,234,174,216)(157,235,175,217)(158,231,171,218)(159,232,172,219)(160,233,173,220), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,193,36)(12,194,37)(13,195,38)(14,191,39)(15,192,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,203,233)(22,204,234)(23,205,235)(24,201,231)(25,202,232)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,228,221)(47,229,222)(48,230,223)(49,226,224)(50,227,225)(71,129,123)(72,130,124)(73,126,125)(74,127,121)(75,128,122)(81,111,150)(82,112,146)(83,113,147)(84,114,148)(85,115,149)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97)(101,136,108)(102,137,109)(103,138,110)(104,139,106)(105,140,107)(131,189,183)(132,190,184)(133,186,185)(134,187,181)(135,188,182)(141,171,210)(142,172,206)(143,173,207)(144,174,208)(145,175,209)(151,178,158)(152,179,159)(153,180,160)(154,176,156)(155,177,157)(161,196,168)(162,197,169)(163,198,170)(164,199,166)(165,200,167)(211,238,218)(212,239,219)(213,240,220)(214,236,216)(215,237,217), (6,45)(7,41)(8,42)(9,43)(10,44)(11,24)(12,25)(13,21)(14,22)(15,23)(16,61)(17,62)(18,63)(19,64)(20,65)(31,88)(32,89)(33,90)(34,86)(35,87)(36,201)(37,202)(38,203)(39,204)(40,205)(46,214)(47,215)(48,211)(49,212)(50,213)(51,77)(52,78)(53,79)(54,80)(55,76)(71,96)(72,97)(73,98)(74,99)(75,100)(81,103)(82,104)(83,105)(84,101)(85,102)(91,125)(92,121)(93,122)(94,123)(95,124)(106,112)(107,113)(108,114)(109,115)(110,111)(116,129)(117,130)(118,126)(119,127)(120,128)(131,196)(132,197)(133,198)(134,199)(135,200)(136,148)(137,149)(138,150)(139,146)(140,147)(141,178)(142,179)(143,180)(144,176)(145,177)(151,171)(152,172)(153,173)(154,174)(155,175)(156,208)(157,209)(158,210)(159,206)(160,207)(161,189)(162,190)(163,186)(164,187)(165,188)(166,181)(167,182)(168,183)(169,184)(170,185)(191,234)(192,235)(193,231)(194,232)(195,233)(216,228)(217,229)(218,230)(219,226)(220,227)(221,236)(222,237)(223,238)(224,239)(225,240) );
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)], [(1,186,66,141),(2,187,67,142),(3,188,68,143),(4,189,69,144),(5,190,70,145),(6,170,52,151),(7,166,53,152),(8,167,54,153),(9,168,55,154),(10,169,51,155),(11,125,231,111),(12,121,232,112),(13,122,233,113),(14,123,234,114),(15,124,235,115),(16,206,86,134),(17,207,87,135),(18,208,88,131),(19,209,89,132),(20,210,90,133),(21,147,195,75),(22,148,191,71),(23,149,192,72),(24,150,193,73),(25,146,194,74),(26,178,58,163),(27,179,59,164),(28,180,60,165),(29,176,56,161),(30,177,57,162),(31,183,63,174),(32,184,64,175),(33,185,65,171),(34,181,61,172),(35,182,62,173),(36,126,201,81),(37,127,202,82),(38,128,203,83),(39,129,204,84),(40,130,205,85),(41,199,79,159),(42,200,80,160),(43,196,76,156),(44,197,77,157),(45,198,78,158),(46,136,216,96),(47,137,217,97),(48,138,218,98),(49,139,219,99),(50,140,220,100),(91,230,110,211),(92,226,106,212),(93,227,107,213),(94,228,108,214),(95,229,109,215),(101,236,116,221),(102,237,117,222),(103,238,118,223),(104,239,119,224),(105,240,120,225)], [(1,118,58,81),(2,119,59,82),(3,120,60,83),(4,116,56,84),(5,117,57,85),(6,150,90,91),(7,146,86,92),(8,147,87,93),(9,148,88,94),(10,149,89,95),(11,185,48,198),(12,181,49,199),(13,182,50,200),(14,183,46,196),(15,184,47,197),(16,106,53,74),(17,107,54,75),(18,108,55,71),(19,109,51,72),(20,110,52,73),(21,207,213,153),(22,208,214,154),(23,209,215,155),(24,210,211,151),(25,206,212,152),(26,126,66,103),(27,127,67,104),(28,128,68,105),(29,129,69,101),(30,130,70,102),(31,96,43,114),(32,97,44,115),(33,98,45,111),(34,99,41,112),(35,100,42,113),(36,186,223,163),(37,187,224,164),(38,188,225,165),(39,189,221,161),(40,190,222,162),(61,139,79,121),(62,140,80,122),(63,136,76,123),(64,137,77,124),(65,138,78,125),(131,228,168,191),(132,229,169,192),(133,230,170,193),(134,226,166,194),(135,227,167,195),(141,238,178,201),(142,239,179,202),(143,240,180,203),(144,236,176,204),(145,237,177,205),(156,234,174,216),(157,235,175,217),(158,231,171,218),(159,232,172,219),(160,233,173,220)], [(1,33,90),(2,34,86),(3,35,87),(4,31,88),(5,32,89),(6,58,45),(7,59,41),(8,60,42),(9,56,43),(10,57,44),(11,193,36),(12,194,37),(13,195,38),(14,191,39),(15,192,40),(16,67,61),(17,68,62),(18,69,63),(19,70,64),(20,66,65),(21,203,233),(22,204,234),(23,205,235),(24,201,231),(25,202,232),(26,78,52),(27,79,53),(28,80,54),(29,76,55),(30,77,51),(46,228,221),(47,229,222),(48,230,223),(49,226,224),(50,227,225),(71,129,123),(72,130,124),(73,126,125),(74,127,121),(75,128,122),(81,111,150),(82,112,146),(83,113,147),(84,114,148),(85,115,149),(91,118,98),(92,119,99),(93,120,100),(94,116,96),(95,117,97),(101,136,108),(102,137,109),(103,138,110),(104,139,106),(105,140,107),(131,189,183),(132,190,184),(133,186,185),(134,187,181),(135,188,182),(141,171,210),(142,172,206),(143,173,207),(144,174,208),(145,175,209),(151,178,158),(152,179,159),(153,180,160),(154,176,156),(155,177,157),(161,196,168),(162,197,169),(163,198,170),(164,199,166),(165,200,167),(211,238,218),(212,239,219),(213,240,220),(214,236,216),(215,237,217)], [(6,45),(7,41),(8,42),(9,43),(10,44),(11,24),(12,25),(13,21),(14,22),(15,23),(16,61),(17,62),(18,63),(19,64),(20,65),(31,88),(32,89),(33,90),(34,86),(35,87),(36,201),(37,202),(38,203),(39,204),(40,205),(46,214),(47,215),(48,211),(49,212),(50,213),(51,77),(52,78),(53,79),(54,80),(55,76),(71,96),(72,97),(73,98),(74,99),(75,100),(81,103),(82,104),(83,105),(84,101),(85,102),(91,125),(92,121),(93,122),(94,123),(95,124),(106,112),(107,113),(108,114),(109,115),(110,111),(116,129),(117,130),(118,126),(119,127),(120,128),(131,196),(132,197),(133,198),(134,199),(135,200),(136,148),(137,149),(138,150),(139,146),(140,147),(141,178),(142,179),(143,180),(144,176),(145,177),(151,171),(152,172),(153,173),(154,174),(155,175),(156,208),(157,209),(158,210),(159,206),(160,207),(161,189),(162,190),(163,186),(164,187),(165,188),(166,181),(167,182),(168,183),(169,184),(170,185),(191,234),(192,235),(193,231),(194,232),(195,233),(216,228),(217,229),(218,230),(219,226),(220,227),(221,236),(222,237),(223,238),(224,239),(225,240)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 20Y | ··· | 20AJ | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D6 | C4○D4 | C5×S3 | C4○D12 | S3×C10 | C5×C4○D4 | C5×C4○D12 |
| kernel | C5×C42⋊3S3 | C5×Dic3⋊C4 | C5×D6⋊C4 | C4×C60 | C42⋊3S3 | Dic3⋊C4 | D6⋊C4 | C4×C12 | C4×C20 | C2×C20 | C30 | C42 | C10 | C2×C4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 3 | 6 | 4 | 12 | 12 | 24 | 48 |
Matrix representation of C5×C42⋊3S3 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 43 | 52 |
| 9 | 43 | 0 | 0 |
| 18 | 52 | 0 | 0 |
| 0 | 0 | 23 | 46 |
| 0 | 0 | 15 | 38 |
| 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 60 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[50,0,0,0,0,50,0,0,0,0,9,43,0,0,18,52],[9,18,0,0,43,52,0,0,0,0,23,15,0,0,46,38],[60,60,0,0,1,0,0,0,0,0,0,60,0,0,1,60],[1,0,0,0,60,60,0,0,0,0,1,60,0,0,0,60] >;
C5×C42⋊3S3 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_3S_3
% in TeX
G:=Group("C5xC4^2:3S3"); // GroupNames label
G:=SmallGroup(480,755);
// by ID
G=gap.SmallGroup(480,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,288,2606,436,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=d^-1>;
// generators/relations