direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C42⋊2S3, (C4×C20)⋊3S3, (C4×S3)⋊3C20, (C4×C12)⋊9C10, (C4×C60)⋊23C2, C42⋊2(C5×S3), (S3×C20)⋊11C4, C4.22(S3×C20), D6⋊C4.7C10, D6.3(C2×C20), C20.124(C4×S3), C12.26(C2×C20), C60.222(C2×C4), (C4×Dic3)⋊8C10, (C2×C20).373D6, C6.3(C22×C20), Dic3⋊C4⋊17C10, (Dic3×C20)⋊26C2, Dic3.5(C2×C20), C15⋊23(C42⋊C2), C30.199(C4○D4), (C2×C30).392C23, C30.194(C22×C4), (C2×C60).451C22, C10.110(C4○D12), (C10×Dic3).212C22, C2.5(S3×C2×C20), (S3×C2×C4).8C10, C6.3(C5×C4○D4), (S3×C2×C20).21C2, C10.130(S3×C2×C4), C3⋊1(C5×C42⋊C2), C2.2(C5×C4○D12), (C2×C4).64(S3×C10), (C5×D6⋊C4).17C2, C22.10(S3×C2×C10), (S3×C10).39(C2×C4), (C5×Dic3⋊C4)⋊39C2, (C2×C12).115(C2×C10), (S3×C2×C10).105C22, (C2×C6).13(C22×C10), (C5×Dic3).47(C2×C4), (C22×S3).14(C2×C10), (C2×C10).326(C22×S3), (C2×Dic3).18(C2×C10), SmallGroup(480,751)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊2S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 308 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C30, C42⋊C2, C2×C20, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C42⋊2S3, S3×C20, C10×Dic3, C10×Dic3, C2×C60, C2×C60, S3×C2×C10, C5×C42⋊C2, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C4×C60, S3×C2×C20, C5×C42⋊2S3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C42⋊C2, C2×C20, C22×C10, S3×C2×C4, C4○D12, S3×C10, C22×C20, C5×C4○D4, C42⋊2S3, S3×C20, S3×C2×C10, C5×C42⋊C2, S3×C2×C20, C5×C4○D12, C5×C42⋊2S3
►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)
(1 178 58 141)(2 179 59 142)(3 180 60 143)(4 176 56 144)(5 177 57 145)(6 210 90 151)(7 206 86 152)(8 207 87 153)(9 208 88 154)(10 209 89 155)(11 125 48 138)(12 121 49 139)(13 122 50 140)(14 123 46 136)(15 124 47 137)(16 166 53 134)(17 167 54 135)(18 168 55 131)(19 169 51 132)(20 170 52 133)(21 147 213 93)(22 148 214 94)(23 149 215 95)(24 150 211 91)(25 146 212 92)(26 186 66 163)(27 187 67 164)(28 188 68 165)(29 189 69 161)(30 190 70 162)(31 156 43 174)(32 157 44 175)(33 158 45 171)(34 159 41 172)(35 160 42 173)(36 126 223 103)(37 127 224 104)(38 128 225 105)(39 129 221 101)(40 130 222 102)(61 199 79 181)(62 200 80 182)(63 196 76 183)(64 197 77 184)(65 198 78 185)(71 228 108 191)(72 229 109 192)(73 230 110 193)(74 226 106 194)(75 227 107 195)(81 238 118 201)(82 239 119 202)(83 240 120 203)(84 236 116 204)(85 237 117 205)(96 234 114 216)(97 235 115 217)(98 231 111 218)(99 232 112 219)(100 233 113 220)
(1 126 66 81)(2 127 67 82)(3 128 68 83)(4 129 69 84)(5 130 70 85)(6 110 52 91)(7 106 53 92)(8 107 54 93)(9 108 55 94)(10 109 51 95)(11 185 231 171)(12 181 232 172)(13 182 233 173)(14 183 234 174)(15 184 235 175)(16 146 86 74)(17 147 87 75)(18 148 88 71)(19 149 89 72)(20 150 90 73)(21 207 195 135)(22 208 191 131)(23 209 192 132)(24 210 193 133)(25 206 194 134)(26 118 58 103)(27 119 59 104)(28 120 60 105)(29 116 56 101)(30 117 57 102)(31 123 63 114)(32 124 64 115)(33 125 65 111)(34 121 61 112)(35 122 62 113)(36 186 201 141)(37 187 202 142)(38 188 203 143)(39 189 204 144)(40 190 205 145)(41 139 79 99)(42 140 80 100)(43 136 76 96)(44 137 77 97)(45 138 78 98)(46 196 216 156)(47 197 217 157)(48 198 218 158)(49 199 219 159)(50 200 220 160)(151 230 170 211)(152 226 166 212)(153 227 167 213)(154 228 168 214)(155 229 169 215)(161 236 176 221)(162 237 177 222)(163 238 178 223)(164 239 179 224)(165 240 180 225)
(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)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 78)(7 79)(8 80)(9 76)(10 77)(11 211)(12 212)(13 213)(14 214)(15 215)(16 34)(17 35)(18 31)(19 32)(20 33)(21 50)(22 46)(23 47)(24 48)(25 49)(26 58)(27 59)(28 60)(29 56)(30 57)(36 238)(37 239)(38 240)(39 236)(40 237)(41 53)(42 54)(43 55)(44 51)(45 52)(61 86)(62 87)(63 88)(64 89)(65 90)(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)(131 174)(132 175)(133 171)(134 172)(135 173)(136 148)(137 149)(138 150)(139 146)(140 147)(141 186)(142 187)(143 188)(144 189)(145 190)(151 198)(152 199)(153 200)(154 196)(155 197)(156 168)(157 169)(158 170)(159 166)(160 167)(161 176)(162 177)(163 178)(164 179)(165 180)(181 206)(182 207)(183 208)(184 209)(185 210)(191 216)(192 217)(193 218)(194 219)(195 220)(201 223)(202 224)(203 225)(204 221)(205 222)(226 232)(227 233)(228 234)(229 235)(230 231)
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), (1,178,58,141)(2,179,59,142)(3,180,60,143)(4,176,56,144)(5,177,57,145)(6,210,90,151)(7,206,86,152)(8,207,87,153)(9,208,88,154)(10,209,89,155)(11,125,48,138)(12,121,49,139)(13,122,50,140)(14,123,46,136)(15,124,47,137)(16,166,53,134)(17,167,54,135)(18,168,55,131)(19,169,51,132)(20,170,52,133)(21,147,213,93)(22,148,214,94)(23,149,215,95)(24,150,211,91)(25,146,212,92)(26,186,66,163)(27,187,67,164)(28,188,68,165)(29,189,69,161)(30,190,70,162)(31,156,43,174)(32,157,44,175)(33,158,45,171)(34,159,41,172)(35,160,42,173)(36,126,223,103)(37,127,224,104)(38,128,225,105)(39,129,221,101)(40,130,222,102)(61,199,79,181)(62,200,80,182)(63,196,76,183)(64,197,77,184)(65,198,78,185)(71,228,108,191)(72,229,109,192)(73,230,110,193)(74,226,106,194)(75,227,107,195)(81,238,118,201)(82,239,119,202)(83,240,120,203)(84,236,116,204)(85,237,117,205)(96,234,114,216)(97,235,115,217)(98,231,111,218)(99,232,112,219)(100,233,113,220), (1,126,66,81)(2,127,67,82)(3,128,68,83)(4,129,69,84)(5,130,70,85)(6,110,52,91)(7,106,53,92)(8,107,54,93)(9,108,55,94)(10,109,51,95)(11,185,231,171)(12,181,232,172)(13,182,233,173)(14,183,234,174)(15,184,235,175)(16,146,86,74)(17,147,87,75)(18,148,88,71)(19,149,89,72)(20,150,90,73)(21,207,195,135)(22,208,191,131)(23,209,192,132)(24,210,193,133)(25,206,194,134)(26,118,58,103)(27,119,59,104)(28,120,60,105)(29,116,56,101)(30,117,57,102)(31,123,63,114)(32,124,64,115)(33,125,65,111)(34,121,61,112)(35,122,62,113)(36,186,201,141)(37,187,202,142)(38,188,203,143)(39,189,204,144)(40,190,205,145)(41,139,79,99)(42,140,80,100)(43,136,76,96)(44,137,77,97)(45,138,78,98)(46,196,216,156)(47,197,217,157)(48,198,218,158)(49,199,219,159)(50,200,220,160)(151,230,170,211)(152,226,166,212)(153,227,167,213)(154,228,168,214)(155,229,169,215)(161,236,176,221)(162,237,177,222)(163,238,178,223)(164,239,179,224)(165,240,180,225), (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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,78)(7,79)(8,80)(9,76)(10,77)(11,211)(12,212)(13,213)(14,214)(15,215)(16,34)(17,35)(18,31)(19,32)(20,33)(21,50)(22,46)(23,47)(24,48)(25,49)(26,58)(27,59)(28,60)(29,56)(30,57)(36,238)(37,239)(38,240)(39,236)(40,237)(41,53)(42,54)(43,55)(44,51)(45,52)(61,86)(62,87)(63,88)(64,89)(65,90)(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)(131,174)(132,175)(133,171)(134,172)(135,173)(136,148)(137,149)(138,150)(139,146)(140,147)(141,186)(142,187)(143,188)(144,189)(145,190)(151,198)(152,199)(153,200)(154,196)(155,197)(156,168)(157,169)(158,170)(159,166)(160,167)(161,176)(162,177)(163,178)(164,179)(165,180)(181,206)(182,207)(183,208)(184,209)(185,210)(191,216)(192,217)(193,218)(194,219)(195,220)(201,223)(202,224)(203,225)(204,221)(205,222)(226,232)(227,233)(228,234)(229,235)(230,231)>;
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), (1,178,58,141)(2,179,59,142)(3,180,60,143)(4,176,56,144)(5,177,57,145)(6,210,90,151)(7,206,86,152)(8,207,87,153)(9,208,88,154)(10,209,89,155)(11,125,48,138)(12,121,49,139)(13,122,50,140)(14,123,46,136)(15,124,47,137)(16,166,53,134)(17,167,54,135)(18,168,55,131)(19,169,51,132)(20,170,52,133)(21,147,213,93)(22,148,214,94)(23,149,215,95)(24,150,211,91)(25,146,212,92)(26,186,66,163)(27,187,67,164)(28,188,68,165)(29,189,69,161)(30,190,70,162)(31,156,43,174)(32,157,44,175)(33,158,45,171)(34,159,41,172)(35,160,42,173)(36,126,223,103)(37,127,224,104)(38,128,225,105)(39,129,221,101)(40,130,222,102)(61,199,79,181)(62,200,80,182)(63,196,76,183)(64,197,77,184)(65,198,78,185)(71,228,108,191)(72,229,109,192)(73,230,110,193)(74,226,106,194)(75,227,107,195)(81,238,118,201)(82,239,119,202)(83,240,120,203)(84,236,116,204)(85,237,117,205)(96,234,114,216)(97,235,115,217)(98,231,111,218)(99,232,112,219)(100,233,113,220), (1,126,66,81)(2,127,67,82)(3,128,68,83)(4,129,69,84)(5,130,70,85)(6,110,52,91)(7,106,53,92)(8,107,54,93)(9,108,55,94)(10,109,51,95)(11,185,231,171)(12,181,232,172)(13,182,233,173)(14,183,234,174)(15,184,235,175)(16,146,86,74)(17,147,87,75)(18,148,88,71)(19,149,89,72)(20,150,90,73)(21,207,195,135)(22,208,191,131)(23,209,192,132)(24,210,193,133)(25,206,194,134)(26,118,58,103)(27,119,59,104)(28,120,60,105)(29,116,56,101)(30,117,57,102)(31,123,63,114)(32,124,64,115)(33,125,65,111)(34,121,61,112)(35,122,62,113)(36,186,201,141)(37,187,202,142)(38,188,203,143)(39,189,204,144)(40,190,205,145)(41,139,79,99)(42,140,80,100)(43,136,76,96)(44,137,77,97)(45,138,78,98)(46,196,216,156)(47,197,217,157)(48,198,218,158)(49,199,219,159)(50,200,220,160)(151,230,170,211)(152,226,166,212)(153,227,167,213)(154,228,168,214)(155,229,169,215)(161,236,176,221)(162,237,177,222)(163,238,178,223)(164,239,179,224)(165,240,180,225), (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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,78)(7,79)(8,80)(9,76)(10,77)(11,211)(12,212)(13,213)(14,214)(15,215)(16,34)(17,35)(18,31)(19,32)(20,33)(21,50)(22,46)(23,47)(24,48)(25,49)(26,58)(27,59)(28,60)(29,56)(30,57)(36,238)(37,239)(38,240)(39,236)(40,237)(41,53)(42,54)(43,55)(44,51)(45,52)(61,86)(62,87)(63,88)(64,89)(65,90)(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)(131,174)(132,175)(133,171)(134,172)(135,173)(136,148)(137,149)(138,150)(139,146)(140,147)(141,186)(142,187)(143,188)(144,189)(145,190)(151,198)(152,199)(153,200)(154,196)(155,197)(156,168)(157,169)(158,170)(159,166)(160,167)(161,176)(162,177)(163,178)(164,179)(165,180)(181,206)(182,207)(183,208)(184,209)(185,210)(191,216)(192,217)(193,218)(194,219)(195,220)(201,223)(202,224)(203,225)(204,221)(205,222)(226,232)(227,233)(228,234)(229,235)(230,231) );
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)], [(1,178,58,141),(2,179,59,142),(3,180,60,143),(4,176,56,144),(5,177,57,145),(6,210,90,151),(7,206,86,152),(8,207,87,153),(9,208,88,154),(10,209,89,155),(11,125,48,138),(12,121,49,139),(13,122,50,140),(14,123,46,136),(15,124,47,137),(16,166,53,134),(17,167,54,135),(18,168,55,131),(19,169,51,132),(20,170,52,133),(21,147,213,93),(22,148,214,94),(23,149,215,95),(24,150,211,91),(25,146,212,92),(26,186,66,163),(27,187,67,164),(28,188,68,165),(29,189,69,161),(30,190,70,162),(31,156,43,174),(32,157,44,175),(33,158,45,171),(34,159,41,172),(35,160,42,173),(36,126,223,103),(37,127,224,104),(38,128,225,105),(39,129,221,101),(40,130,222,102),(61,199,79,181),(62,200,80,182),(63,196,76,183),(64,197,77,184),(65,198,78,185),(71,228,108,191),(72,229,109,192),(73,230,110,193),(74,226,106,194),(75,227,107,195),(81,238,118,201),(82,239,119,202),(83,240,120,203),(84,236,116,204),(85,237,117,205),(96,234,114,216),(97,235,115,217),(98,231,111,218),(99,232,112,219),(100,233,113,220)], [(1,126,66,81),(2,127,67,82),(3,128,68,83),(4,129,69,84),(5,130,70,85),(6,110,52,91),(7,106,53,92),(8,107,54,93),(9,108,55,94),(10,109,51,95),(11,185,231,171),(12,181,232,172),(13,182,233,173),(14,183,234,174),(15,184,235,175),(16,146,86,74),(17,147,87,75),(18,148,88,71),(19,149,89,72),(20,150,90,73),(21,207,195,135),(22,208,191,131),(23,209,192,132),(24,210,193,133),(25,206,194,134),(26,118,58,103),(27,119,59,104),(28,120,60,105),(29,116,56,101),(30,117,57,102),(31,123,63,114),(32,124,64,115),(33,125,65,111),(34,121,61,112),(35,122,62,113),(36,186,201,141),(37,187,202,142),(38,188,203,143),(39,189,204,144),(40,190,205,145),(41,139,79,99),(42,140,80,100),(43,136,76,96),(44,137,77,97),(45,138,78,98),(46,196,216,156),(47,197,217,157),(48,198,218,158),(49,199,219,159),(50,200,220,160),(151,230,170,211),(152,226,166,212),(153,227,167,213),(154,228,168,214),(155,229,169,215),(161,236,176,221),(162,237,177,222),(163,238,178,223),(164,239,179,224),(165,240,180,225)], [(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)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,78),(7,79),(8,80),(9,76),(10,77),(11,211),(12,212),(13,213),(14,214),(15,215),(16,34),(17,35),(18,31),(19,32),(20,33),(21,50),(22,46),(23,47),(24,48),(25,49),(26,58),(27,59),(28,60),(29,56),(30,57),(36,238),(37,239),(38,240),(39,236),(40,237),(41,53),(42,54),(43,55),(44,51),(45,52),(61,86),(62,87),(63,88),(64,89),(65,90),(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),(131,174),(132,175),(133,171),(134,172),(135,173),(136,148),(137,149),(138,150),(139,146),(140,147),(141,186),(142,187),(143,188),(144,189),(145,190),(151,198),(152,199),(153,200),(154,196),(155,197),(156,168),(157,169),(158,170),(159,166),(160,167),(161,176),(162,177),(163,178),(164,179),(165,180),(181,206),(182,207),(183,208),(184,209),(185,210),(191,216),(192,217),(193,218),(194,219),(195,220),(201,223),(202,224),(203,225),(204,221),(205,222),(226,232),(227,233),(228,234),(229,235),(230,231)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20BD | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | S3 | D6 | C4○D4 | C4×S3 | C5×S3 | C4○D12 | S3×C10 | C5×C4○D4 | S3×C20 | C5×C4○D12 |
| kernel | C5×C42⋊2S3 | Dic3×C20 | C5×Dic3⋊C4 | C5×D6⋊C4 | C4×C60 | S3×C2×C20 | S3×C20 | C42⋊2S3 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C4×C12 | S3×C2×C4 | C4×S3 | C4×C20 | C2×C20 | C30 | C20 | C42 | C10 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 4 | 4 | 8 | 8 | 4 | 4 | 32 | 1 | 3 | 4 | 4 | 4 | 8 | 12 | 16 | 16 | 32 |
Matrix representation of C5×C42⋊2S3 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 34 | 0 |
| 0 | 0 | 34 |
| 60 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 50 |
| 11 | 0 | 0 |
| 0 | 52 | 43 |
| 0 | 18 | 9 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 60 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 60 | 60 |
G:=sub<GL(3,GF(61))| [1,0,0,0,34,0,0,0,34],[60,0,0,0,50,0,0,0,50],[11,0,0,0,52,18,0,43,9],[1,0,0,0,0,60,0,1,60],[1,0,0,0,1,60,0,0,60] >;
C5×C42⋊2S3 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_2S_3
% in TeX
G:=Group("C5xC4^2:2S3"); // GroupNames label
G:=SmallGroup(480,751);
// by ID
G=gap.SmallGroup(480,751);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,1766,226,15686]);
// 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,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations