direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C23.28D6, D6⋊C4⋊2C10, (C22×C60)⋊4C2, (C22×C20)⋊7S3, C6.42(D4×C10), Dic3⋊C4⋊3C10, (C22×C12)⋊2C10, (C2×C20).377D6, C30.425(C2×D4), (C2×C30).180D4, C6.D4⋊6C10, C23.28(S3×C10), C30.211(C4○D4), (C2×C30).426C23, (C2×C60).457C22, (C22×C10).125D6, C10.125(C4○D12), C15⋊36(C22.D4), (C22×C30).177C22, (C10×Dic3).148C22, (C5×D6⋊C4)⋊2C2, (C22×C4)⋊5(C5×S3), C6.16(C5×C4○D4), (C2×C6).37(C5×D4), C2.6(C10×C3⋊D4), (C2×C4).68(S3×C10), (C5×Dic3⋊C4)⋊3C2, C2.18(C5×C4○D12), (C2×C3⋊D4).6C10, C22.55(S3×C2×C10), C3⋊4(C5×C22.D4), C22.9(C5×C3⋊D4), (C2×C12).76(C2×C10), (C10×C3⋊D4).13C2, C10.127(C2×C3⋊D4), (S3×C2×C10).70C22, (C5×C6.D4)⋊22C2, (C2×C10).62(C3⋊D4), (C22×S3).9(C2×C10), (C22×C6).39(C2×C10), (C2×C6).47(C22×C10), (C2×C10).360(C22×S3), (C2×Dic3).12(C2×C10), SmallGroup(480,808)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.28D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 356 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22.D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C23.28D6, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, C2×C60, S3×C2×C10, C22×C30, C5×C22.D4, C5×Dic3⋊C4, C5×D6⋊C4, C5×C6.D4, C10×C3⋊D4, C22×C60, C5×C23.28D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C22.D4, C5×D4, C22×C10, C4○D12, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C23.28D6, C5×C3⋊D4, S3×C2×C10, C5×C22.D4, C5×C4○D12, C10×C3⋊D4, C5×C23.28D6
►On 240 points
Generators in S240
(1 169 63 100 22)(2 170 64 101 23)(3 171 65 102 24)(4 172 66 103 13)(5 173 67 104 14)(6 174 68 105 15)(7 175 69 106 16)(8 176 70 107 17)(9 177 71 108 18)(10 178 72 97 19)(11 179 61 98 20)(12 180 62 99 21)(25 221 137 197 109)(26 222 138 198 110)(27 223 139 199 111)(28 224 140 200 112)(29 225 141 201 113)(30 226 142 202 114)(31 227 143 203 115)(32 228 144 204 116)(33 217 133 193 117)(34 218 134 194 118)(35 219 135 195 119)(36 220 136 196 120)(37 85 56 183 168)(38 86 57 184 157)(39 87 58 185 158)(40 88 59 186 159)(41 89 60 187 160)(42 90 49 188 161)(43 91 50 189 162)(44 92 51 190 163)(45 93 52 191 164)(46 94 53 192 165)(47 95 54 181 166)(48 96 55 182 167)(73 230 148 123 211)(74 231 149 124 212)(75 232 150 125 213)(76 233 151 126 214)(77 234 152 127 215)(78 235 153 128 216)(79 236 154 129 205)(80 237 155 130 206)(81 238 156 131 207)(82 239 145 132 208)(83 240 146 121 209)(84 229 147 122 210)
(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(1 120)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 199)(14 200)(15 201)(16 202)(17 203)(18 204)(19 193)(20 194)(21 195)(22 196)(23 197)(24 198)(25 170)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 169)(37 150)(38 151)(39 152)(40 153)(41 154)(42 155)(43 156)(44 145)(45 146)(46 147)(47 148)(48 149)(49 206)(50 207)(51 208)(52 209)(53 210)(54 211)(55 212)(56 213)(57 214)(58 215)(59 216)(60 205)(61 218)(62 219)(63 220)(64 221)(65 222)(66 223)(67 224)(68 225)(69 226)(70 227)(71 228)(72 217)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 121)(94 122)(95 123)(96 124)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)(157 233)(158 234)(159 235)(160 236)(161 237)(162 238)(163 239)(164 240)(165 229)(166 230)(167 231)(168 232)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(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)
(1 234 114 164)(2 163 115 233)(3 232 116 162)(4 161 117 231)(5 230 118 160)(6 159 119 229)(7 240 120 158)(8 157 109 239)(9 238 110 168)(10 167 111 237)(11 236 112 166)(12 165 113 235)(13 188 193 74)(14 73 194 187)(15 186 195 84)(16 83 196 185)(17 184 197 82)(18 81 198 183)(19 182 199 80)(20 79 200 181)(21 192 201 78)(22 77 202 191)(23 190 203 76)(24 75 204 189)(25 145 176 38)(26 37 177 156)(27 155 178 48)(28 47 179 154)(29 153 180 46)(30 45 169 152)(31 151 170 44)(32 43 171 150)(33 149 172 42)(34 41 173 148)(35 147 174 40)(36 39 175 146)(49 133 212 103)(50 102 213 144)(51 143 214 101)(52 100 215 142)(53 141 216 99)(54 98 205 140)(55 139 206 97)(56 108 207 138)(57 137 208 107)(58 106 209 136)(59 135 210 105)(60 104 211 134)(61 129 224 95)(62 94 225 128)(63 127 226 93)(64 92 227 126)(65 125 228 91)(66 90 217 124)(67 123 218 89)(68 88 219 122)(69 121 220 87)(70 86 221 132)(71 131 222 85)(72 96 223 130)
G:=sub<Sym(240)| (1,169,63,100,22)(2,170,64,101,23)(3,171,65,102,24)(4,172,66,103,13)(5,173,67,104,14)(6,174,68,105,15)(7,175,69,106,16)(8,176,70,107,17)(9,177,71,108,18)(10,178,72,97,19)(11,179,61,98,20)(12,180,62,99,21)(25,221,137,197,109)(26,222,138,198,110)(27,223,139,199,111)(28,224,140,200,112)(29,225,141,201,113)(30,226,142,202,114)(31,227,143,203,115)(32,228,144,204,116)(33,217,133,193,117)(34,218,134,194,118)(35,219,135,195,119)(36,220,136,196,120)(37,85,56,183,168)(38,86,57,184,157)(39,87,58,185,158)(40,88,59,186,159)(41,89,60,187,160)(42,90,49,188,161)(43,91,50,189,162)(44,92,51,190,163)(45,93,52,191,164)(46,94,53,192,165)(47,95,54,181,166)(48,96,55,182,167)(73,230,148,123,211)(74,231,149,124,212)(75,232,150,125,213)(76,233,151,126,214)(77,234,152,127,215)(78,235,153,128,216)(79,236,154,129,205)(80,237,155,130,206)(81,238,156,131,207)(82,239,145,132,208)(83,240,146,121,209)(84,229,147,122,210), (37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,120)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,199)(14,200)(15,201)(16,202)(17,203)(18,204)(19,193)(20,194)(21,195)(22,196)(23,197)(24,198)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,169)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,145)(45,146)(46,147)(47,148)(48,149)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,212)(56,213)(57,214)(58,215)(59,216)(60,205)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,217)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,121)(94,122)(95,123)(96,124)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,229)(166,230)(167,231)(168,232), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,234,114,164)(2,163,115,233)(3,232,116,162)(4,161,117,231)(5,230,118,160)(6,159,119,229)(7,240,120,158)(8,157,109,239)(9,238,110,168)(10,167,111,237)(11,236,112,166)(12,165,113,235)(13,188,193,74)(14,73,194,187)(15,186,195,84)(16,83,196,185)(17,184,197,82)(18,81,198,183)(19,182,199,80)(20,79,200,181)(21,192,201,78)(22,77,202,191)(23,190,203,76)(24,75,204,189)(25,145,176,38)(26,37,177,156)(27,155,178,48)(28,47,179,154)(29,153,180,46)(30,45,169,152)(31,151,170,44)(32,43,171,150)(33,149,172,42)(34,41,173,148)(35,147,174,40)(36,39,175,146)(49,133,212,103)(50,102,213,144)(51,143,214,101)(52,100,215,142)(53,141,216,99)(54,98,205,140)(55,139,206,97)(56,108,207,138)(57,137,208,107)(58,106,209,136)(59,135,210,105)(60,104,211,134)(61,129,224,95)(62,94,225,128)(63,127,226,93)(64,92,227,126)(65,125,228,91)(66,90,217,124)(67,123,218,89)(68,88,219,122)(69,121,220,87)(70,86,221,132)(71,131,222,85)(72,96,223,130)>;
G:=Group( (1,169,63,100,22)(2,170,64,101,23)(3,171,65,102,24)(4,172,66,103,13)(5,173,67,104,14)(6,174,68,105,15)(7,175,69,106,16)(8,176,70,107,17)(9,177,71,108,18)(10,178,72,97,19)(11,179,61,98,20)(12,180,62,99,21)(25,221,137,197,109)(26,222,138,198,110)(27,223,139,199,111)(28,224,140,200,112)(29,225,141,201,113)(30,226,142,202,114)(31,227,143,203,115)(32,228,144,204,116)(33,217,133,193,117)(34,218,134,194,118)(35,219,135,195,119)(36,220,136,196,120)(37,85,56,183,168)(38,86,57,184,157)(39,87,58,185,158)(40,88,59,186,159)(41,89,60,187,160)(42,90,49,188,161)(43,91,50,189,162)(44,92,51,190,163)(45,93,52,191,164)(46,94,53,192,165)(47,95,54,181,166)(48,96,55,182,167)(73,230,148,123,211)(74,231,149,124,212)(75,232,150,125,213)(76,233,151,126,214)(77,234,152,127,215)(78,235,153,128,216)(79,236,154,129,205)(80,237,155,130,206)(81,238,156,131,207)(82,239,145,132,208)(83,240,146,121,209)(84,229,147,122,210), (37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,120)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,199)(14,200)(15,201)(16,202)(17,203)(18,204)(19,193)(20,194)(21,195)(22,196)(23,197)(24,198)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,169)(37,150)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,145)(45,146)(46,147)(47,148)(48,149)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,212)(56,213)(57,214)(58,215)(59,216)(60,205)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,217)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,121)(94,122)(95,123)(96,124)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,229)(166,230)(167,231)(168,232), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (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), (1,234,114,164)(2,163,115,233)(3,232,116,162)(4,161,117,231)(5,230,118,160)(6,159,119,229)(7,240,120,158)(8,157,109,239)(9,238,110,168)(10,167,111,237)(11,236,112,166)(12,165,113,235)(13,188,193,74)(14,73,194,187)(15,186,195,84)(16,83,196,185)(17,184,197,82)(18,81,198,183)(19,182,199,80)(20,79,200,181)(21,192,201,78)(22,77,202,191)(23,190,203,76)(24,75,204,189)(25,145,176,38)(26,37,177,156)(27,155,178,48)(28,47,179,154)(29,153,180,46)(30,45,169,152)(31,151,170,44)(32,43,171,150)(33,149,172,42)(34,41,173,148)(35,147,174,40)(36,39,175,146)(49,133,212,103)(50,102,213,144)(51,143,214,101)(52,100,215,142)(53,141,216,99)(54,98,205,140)(55,139,206,97)(56,108,207,138)(57,137,208,107)(58,106,209,136)(59,135,210,105)(60,104,211,134)(61,129,224,95)(62,94,225,128)(63,127,226,93)(64,92,227,126)(65,125,228,91)(66,90,217,124)(67,123,218,89)(68,88,219,122)(69,121,220,87)(70,86,221,132)(71,131,222,85)(72,96,223,130) );
G=PermutationGroup([[(1,169,63,100,22),(2,170,64,101,23),(3,171,65,102,24),(4,172,66,103,13),(5,173,67,104,14),(6,174,68,105,15),(7,175,69,106,16),(8,176,70,107,17),(9,177,71,108,18),(10,178,72,97,19),(11,179,61,98,20),(12,180,62,99,21),(25,221,137,197,109),(26,222,138,198,110),(27,223,139,199,111),(28,224,140,200,112),(29,225,141,201,113),(30,226,142,202,114),(31,227,143,203,115),(32,228,144,204,116),(33,217,133,193,117),(34,218,134,194,118),(35,219,135,195,119),(36,220,136,196,120),(37,85,56,183,168),(38,86,57,184,157),(39,87,58,185,158),(40,88,59,186,159),(41,89,60,187,160),(42,90,49,188,161),(43,91,50,189,162),(44,92,51,190,163),(45,93,52,191,164),(46,94,53,192,165),(47,95,54,181,166),(48,96,55,182,167),(73,230,148,123,211),(74,231,149,124,212),(75,232,150,125,213),(76,233,151,126,214),(77,234,152,127,215),(78,235,153,128,216),(79,236,154,129,205),(80,237,155,130,206),(81,238,156,131,207),(82,239,145,132,208),(83,240,146,121,209),(84,229,147,122,210)], [(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(1,120),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,199),(14,200),(15,201),(16,202),(17,203),(18,204),(19,193),(20,194),(21,195),(22,196),(23,197),(24,198),(25,170),(26,171),(27,172),(28,173),(29,174),(30,175),(31,176),(32,177),(33,178),(34,179),(35,180),(36,169),(37,150),(38,151),(39,152),(40,153),(41,154),(42,155),(43,156),(44,145),(45,146),(46,147),(47,148),(48,149),(49,206),(50,207),(51,208),(52,209),(53,210),(54,211),(55,212),(56,213),(57,214),(58,215),(59,216),(60,205),(61,218),(62,219),(63,220),(64,221),(65,222),(66,223),(67,224),(68,225),(69,226),(70,227),(71,228),(72,217),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,189),(82,190),(83,191),(84,192),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,121),(94,122),(95,123),(96,124),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144),(157,233),(158,234),(159,235),(160,236),(161,237),(162,238),(163,239),(164,240),(165,229),(166,230),(167,231),(168,232)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(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)], [(1,234,114,164),(2,163,115,233),(3,232,116,162),(4,161,117,231),(5,230,118,160),(6,159,119,229),(7,240,120,158),(8,157,109,239),(9,238,110,168),(10,167,111,237),(11,236,112,166),(12,165,113,235),(13,188,193,74),(14,73,194,187),(15,186,195,84),(16,83,196,185),(17,184,197,82),(18,81,198,183),(19,182,199,80),(20,79,200,181),(21,192,201,78),(22,77,202,191),(23,190,203,76),(24,75,204,189),(25,145,176,38),(26,37,177,156),(27,155,178,48),(28,47,179,154),(29,153,180,46),(30,45,169,152),(31,151,170,44),(32,43,171,150),(33,149,172,42),(34,41,173,148),(35,147,174,40),(36,39,175,146),(49,133,212,103),(50,102,213,144),(51,143,214,101),(52,100,215,142),(53,141,216,99),(54,98,205,140),(55,139,206,97),(56,108,207,138),(57,137,208,107),(58,106,209,136),(59,135,210,105),(60,104,211,134),(61,129,224,95),(62,94,225,128),(63,127,226,93),(64,92,227,126),(65,125,228,91),(66,90,217,124),(67,123,218,89),(68,88,219,122),(69,121,220,87),(70,86,221,132),(71,131,222,85),(72,96,223,130)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AB | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 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 | 2 | 2 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C5×S3 | C5×D4 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | C5×C4○D12 |
| kernel | C5×C23.28D6 | C5×Dic3⋊C4 | C5×D6⋊C4 | C5×C6.D4 | C10×C3⋊D4 | C22×C60 | C23.28D6 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22×C12 | C22×C20 | C2×C30 | C2×C20 | C22×C10 | C30 | C2×C10 | C22×C4 | C2×C6 | C10 | C2×C4 | C23 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C5×C23.28D6 ►in GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 1 | 0 | 0 | 0 |
| 30 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 29 | 0 |
| 0 | 0 | 57 | 40 |
| 7 | 28 | 0 | 0 |
| 20 | 54 | 0 | 0 |
| 0 | 0 | 50 | 15 |
| 0 | 0 | 53 | 11 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,20,0,0,0,0,20],[1,30,0,0,0,60,0,0,0,0,1,34,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[50,0,0,0,0,50,0,0,0,0,29,57,0,0,0,40],[7,20,0,0,28,54,0,0,0,0,50,53,0,0,15,11] >;
C5×C23.28D6 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{28}D_6 % in TeX
G:=Group("C5xC2^3.28D6"); // GroupNames label
G:=SmallGroup(480,808);
// by ID
G=gap.SmallGroup(480,808);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,436,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations