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G = C5×C23.26D6order 480 = 25·3·5

Direct product of C5 and C23.26D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C23.26D6, (C2×C60)⋊25C4, (C2×C12)⋊6C20, C60.256(C2×C4), C12.42(C2×C20), C4⋊Dic317C10, (C2×C20)⋊14Dic3, (C2×C20).439D6, (C4×Dic3)⋊15C10, (Dic3×C20)⋊33C2, C6.24(C22×C20), (C22×C60).26C2, (C22×C20).21S3, C23.26(S3×C10), C20.70(C2×Dic3), C4.15(C10×Dic3), C1532(C42⋊C2), C30.209(C4○D4), (C2×C30).423C23, (C22×C12).10C10, C30.231(C22×C4), (C2×C60).568C22, C6.D4.5C10, (C22×C10).123D6, C10.123(C4○D12), C22.5(C10×Dic3), C10.47(C22×Dic3), (C22×C30).174C22, (C10×Dic3).225C22, C34(C5×C42⋊C2), (C2×C4)⋊4(C5×Dic3), C6.14(C5×C4○D4), C2.4(C5×C4○D12), C2.5(Dic3×C2×C10), (C2×C6).35(C2×C20), (C5×C4⋊Dic3)⋊35C2, (C22×C4).9(C5×S3), C22.22(S3×C2×C10), (C2×C4).102(S3×C10), (C2×C30).203(C2×C4), (C2×C12).98(C2×C10), (C2×C6).44(C22×C10), (C22×C6).36(C2×C10), (C2×C10).44(C2×Dic3), (C2×C10).357(C22×S3), (C2×Dic3).33(C2×C10), (C5×C6.D4).11C2, SmallGroup(480,805)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C23.26D6
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C20 — C5×C23.26D6
C3C6 — C5×C23.26D6
C1C2×C20C22×C20

Generators and relations for C5×C23.26D6
 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=e5 >

Subgroups: 260 in 152 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, C5×Dic3 [×4], C60 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, C23.26D6, C10×Dic3 [×4], C2×C60 [×2], C2×C60 [×4], C22×C30, C5×C42⋊C2, Dic3×C20 [×2], C5×C4⋊Dic3 [×2], C5×C6.D4 [×2], C22×C60, C5×C23.26D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×S3, C5×S3, C42⋊C2, C2×C20 [×6], C22×C10, C4○D12 [×2], C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C22×C20, C5×C4○D4 [×2], C23.26D6, C10×Dic3 [×6], S3×C2×C10, C5×C42⋊C2, C5×C4○D12 [×2], Dic3×C2×C10, C5×C23.26D6

Smallest permutation representation of C5×C23.26D6
On 240 points
Generators in S240
(1 197 15 91 143)(2 198 16 92 144)(3 199 17 93 133)(4 200 18 94 134)(5 201 19 95 135)(6 202 20 96 136)(7 203 21 85 137)(8 204 22 86 138)(9 193 23 87 139)(10 194 24 88 140)(11 195 13 89 141)(12 196 14 90 142)(25 205 123 101 111)(26 206 124 102 112)(27 207 125 103 113)(28 208 126 104 114)(29 209 127 105 115)(30 210 128 106 116)(31 211 129 107 117)(32 212 130 108 118)(33 213 131 97 119)(34 214 132 98 120)(35 215 121 99 109)(36 216 122 100 110)(37 82 166 153 63)(38 83 167 154 64)(39 84 168 155 65)(40 73 157 156 66)(41 74 158 145 67)(42 75 159 146 68)(43 76 160 147 69)(44 77 161 148 70)(45 78 162 149 71)(46 79 163 150 72)(47 80 164 151 61)(48 81 165 152 62)(49 233 181 180 222)(50 234 182 169 223)(51 235 183 170 224)(52 236 184 171 225)(53 237 185 172 226)(54 238 186 173 227)(55 239 187 174 228)(56 240 188 175 217)(57 229 189 176 218)(58 230 190 177 219)(59 231 191 178 220)(60 232 192 179 221)
(1 208)(2 209)(3 210)(4 211)(5 212)(6 213)(7 214)(8 215)(9 216)(10 205)(11 206)(12 207)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 97)(21 98)(22 99)(23 100)(24 101)(25 140)(26 141)(27 142)(28 143)(29 144)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 175)(38 176)(39 177)(40 178)(41 179)(42 180)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 159)(50 160)(51 161)(52 162)(53 163)(54 164)(55 165)(56 166)(57 167)(58 168)(59 157)(60 158)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 181)(69 182)(70 183)(71 184)(72 185)(73 220)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 228)(82 217)(83 218)(84 219)(85 120)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(121 204)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(145 232)(146 233)(147 234)(148 235)(149 236)(150 237)(151 238)(152 239)(153 240)(154 229)(155 230)(156 231)
(1 208)(2 209)(3 210)(4 211)(5 212)(6 213)(7 214)(8 215)(9 216)(10 205)(11 206)(12 207)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 97)(21 98)(22 99)(23 100)(24 101)(25 140)(26 141)(27 142)(28 143)(29 144)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 169)(38 170)(39 171)(40 172)(41 173)(42 174)(43 175)(44 176)(45 177)(46 178)(47 179)(48 180)(49 165)(50 166)(51 167)(52 168)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 192)(62 181)(63 182)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(71 190)(72 191)(73 226)(74 227)(75 228)(76 217)(77 218)(78 219)(79 220)(80 221)(81 222)(82 223)(83 224)(84 225)(85 120)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(121 204)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(145 238)(146 239)(147 240)(148 229)(149 230)(150 231)(151 232)(152 233)(153 234)(154 235)(155 236)(156 237)
(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)
(1 188 214 63)(2 181 215 68)(3 186 216 61)(4 191 205 66)(5 184 206 71)(6 189 207 64)(7 182 208 69)(8 187 209 62)(9 192 210 67)(10 185 211 72)(11 190 212 65)(12 183 213 70)(13 219 108 84)(14 224 97 77)(15 217 98 82)(16 222 99 75)(17 227 100 80)(18 220 101 73)(19 225 102 78)(20 218 103 83)(21 223 104 76)(22 228 105 81)(23 221 106 74)(24 226 107 79)(25 156 134 231)(26 149 135 236)(27 154 136 229)(28 147 137 234)(29 152 138 239)(30 145 139 232)(31 150 140 237)(32 155 141 230)(33 148 142 235)(34 153 143 240)(35 146 144 233)(36 151 133 238)(37 197 175 132)(38 202 176 125)(39 195 177 130)(40 200 178 123)(41 193 179 128)(42 198 180 121)(43 203 169 126)(44 196 170 131)(45 201 171 124)(46 194 172 129)(47 199 173 122)(48 204 174 127)(49 109 159 92)(50 114 160 85)(51 119 161 90)(52 112 162 95)(53 117 163 88)(54 110 164 93)(55 115 165 86)(56 120 166 91)(57 113 167 96)(58 118 168 89)(59 111 157 94)(60 116 158 87)

G:=sub<Sym(240)| (1,197,15,91,143)(2,198,16,92,144)(3,199,17,93,133)(4,200,18,94,134)(5,201,19,95,135)(6,202,20,96,136)(7,203,21,85,137)(8,204,22,86,138)(9,193,23,87,139)(10,194,24,88,140)(11,195,13,89,141)(12,196,14,90,142)(25,205,123,101,111)(26,206,124,102,112)(27,207,125,103,113)(28,208,126,104,114)(29,209,127,105,115)(30,210,128,106,116)(31,211,129,107,117)(32,212,130,108,118)(33,213,131,97,119)(34,214,132,98,120)(35,215,121,99,109)(36,216,122,100,110)(37,82,166,153,63)(38,83,167,154,64)(39,84,168,155,65)(40,73,157,156,66)(41,74,158,145,67)(42,75,159,146,68)(43,76,160,147,69)(44,77,161,148,70)(45,78,162,149,71)(46,79,163,150,72)(47,80,164,151,61)(48,81,165,152,62)(49,233,181,180,222)(50,234,182,169,223)(51,235,183,170,224)(52,236,184,171,225)(53,237,185,172,226)(54,238,186,173,227)(55,239,187,174,228)(56,240,188,175,217)(57,229,189,176,218)(58,230,190,177,219)(59,231,191,178,220)(60,232,192,179,221), (1,208)(2,209)(3,210)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,205)(11,206)(12,207)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,97)(21,98)(22,99)(23,100)(24,101)(25,140)(26,141)(27,142)(28,143)(29,144)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,175)(38,176)(39,177)(40,178)(41,179)(42,180)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,159)(50,160)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,167)(58,168)(59,157)(60,158)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,181)(69,182)(70,183)(71,184)(72,185)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,217)(83,218)(84,219)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,204)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,229)(155,230)(156,231), (1,208)(2,209)(3,210)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,205)(11,206)(12,207)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,97)(21,98)(22,99)(23,100)(24,101)(25,140)(26,141)(27,142)(28,143)(29,144)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,192)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,226)(74,227)(75,228)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,225)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,204)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(145,238)(146,239)(147,240)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237), (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), (1,188,214,63)(2,181,215,68)(3,186,216,61)(4,191,205,66)(5,184,206,71)(6,189,207,64)(7,182,208,69)(8,187,209,62)(9,192,210,67)(10,185,211,72)(11,190,212,65)(12,183,213,70)(13,219,108,84)(14,224,97,77)(15,217,98,82)(16,222,99,75)(17,227,100,80)(18,220,101,73)(19,225,102,78)(20,218,103,83)(21,223,104,76)(22,228,105,81)(23,221,106,74)(24,226,107,79)(25,156,134,231)(26,149,135,236)(27,154,136,229)(28,147,137,234)(29,152,138,239)(30,145,139,232)(31,150,140,237)(32,155,141,230)(33,148,142,235)(34,153,143,240)(35,146,144,233)(36,151,133,238)(37,197,175,132)(38,202,176,125)(39,195,177,130)(40,200,178,123)(41,193,179,128)(42,198,180,121)(43,203,169,126)(44,196,170,131)(45,201,171,124)(46,194,172,129)(47,199,173,122)(48,204,174,127)(49,109,159,92)(50,114,160,85)(51,119,161,90)(52,112,162,95)(53,117,163,88)(54,110,164,93)(55,115,165,86)(56,120,166,91)(57,113,167,96)(58,118,168,89)(59,111,157,94)(60,116,158,87)>;

G:=Group( (1,197,15,91,143)(2,198,16,92,144)(3,199,17,93,133)(4,200,18,94,134)(5,201,19,95,135)(6,202,20,96,136)(7,203,21,85,137)(8,204,22,86,138)(9,193,23,87,139)(10,194,24,88,140)(11,195,13,89,141)(12,196,14,90,142)(25,205,123,101,111)(26,206,124,102,112)(27,207,125,103,113)(28,208,126,104,114)(29,209,127,105,115)(30,210,128,106,116)(31,211,129,107,117)(32,212,130,108,118)(33,213,131,97,119)(34,214,132,98,120)(35,215,121,99,109)(36,216,122,100,110)(37,82,166,153,63)(38,83,167,154,64)(39,84,168,155,65)(40,73,157,156,66)(41,74,158,145,67)(42,75,159,146,68)(43,76,160,147,69)(44,77,161,148,70)(45,78,162,149,71)(46,79,163,150,72)(47,80,164,151,61)(48,81,165,152,62)(49,233,181,180,222)(50,234,182,169,223)(51,235,183,170,224)(52,236,184,171,225)(53,237,185,172,226)(54,238,186,173,227)(55,239,187,174,228)(56,240,188,175,217)(57,229,189,176,218)(58,230,190,177,219)(59,231,191,178,220)(60,232,192,179,221), (1,208)(2,209)(3,210)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,205)(11,206)(12,207)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,97)(21,98)(22,99)(23,100)(24,101)(25,140)(26,141)(27,142)(28,143)(29,144)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,175)(38,176)(39,177)(40,178)(41,179)(42,180)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,159)(50,160)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,167)(58,168)(59,157)(60,158)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,181)(69,182)(70,183)(71,184)(72,185)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,217)(83,218)(84,219)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,204)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,229)(155,230)(156,231), (1,208)(2,209)(3,210)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,205)(11,206)(12,207)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,97)(21,98)(22,99)(23,100)(24,101)(25,140)(26,141)(27,142)(28,143)(29,144)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,179)(48,180)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,192)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,226)(74,227)(75,228)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,225)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,204)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(145,238)(146,239)(147,240)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237), (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), (1,188,214,63)(2,181,215,68)(3,186,216,61)(4,191,205,66)(5,184,206,71)(6,189,207,64)(7,182,208,69)(8,187,209,62)(9,192,210,67)(10,185,211,72)(11,190,212,65)(12,183,213,70)(13,219,108,84)(14,224,97,77)(15,217,98,82)(16,222,99,75)(17,227,100,80)(18,220,101,73)(19,225,102,78)(20,218,103,83)(21,223,104,76)(22,228,105,81)(23,221,106,74)(24,226,107,79)(25,156,134,231)(26,149,135,236)(27,154,136,229)(28,147,137,234)(29,152,138,239)(30,145,139,232)(31,150,140,237)(32,155,141,230)(33,148,142,235)(34,153,143,240)(35,146,144,233)(36,151,133,238)(37,197,175,132)(38,202,176,125)(39,195,177,130)(40,200,178,123)(41,193,179,128)(42,198,180,121)(43,203,169,126)(44,196,170,131)(45,201,171,124)(46,194,172,129)(47,199,173,122)(48,204,174,127)(49,109,159,92)(50,114,160,85)(51,119,161,90)(52,112,162,95)(53,117,163,88)(54,110,164,93)(55,115,165,86)(56,120,166,91)(57,113,167,96)(58,118,168,89)(59,111,157,94)(60,116,158,87) );

G=PermutationGroup([(1,197,15,91,143),(2,198,16,92,144),(3,199,17,93,133),(4,200,18,94,134),(5,201,19,95,135),(6,202,20,96,136),(7,203,21,85,137),(8,204,22,86,138),(9,193,23,87,139),(10,194,24,88,140),(11,195,13,89,141),(12,196,14,90,142),(25,205,123,101,111),(26,206,124,102,112),(27,207,125,103,113),(28,208,126,104,114),(29,209,127,105,115),(30,210,128,106,116),(31,211,129,107,117),(32,212,130,108,118),(33,213,131,97,119),(34,214,132,98,120),(35,215,121,99,109),(36,216,122,100,110),(37,82,166,153,63),(38,83,167,154,64),(39,84,168,155,65),(40,73,157,156,66),(41,74,158,145,67),(42,75,159,146,68),(43,76,160,147,69),(44,77,161,148,70),(45,78,162,149,71),(46,79,163,150,72),(47,80,164,151,61),(48,81,165,152,62),(49,233,181,180,222),(50,234,182,169,223),(51,235,183,170,224),(52,236,184,171,225),(53,237,185,172,226),(54,238,186,173,227),(55,239,187,174,228),(56,240,188,175,217),(57,229,189,176,218),(58,230,190,177,219),(59,231,191,178,220),(60,232,192,179,221)], [(1,208),(2,209),(3,210),(4,211),(5,212),(6,213),(7,214),(8,215),(9,216),(10,205),(11,206),(12,207),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,97),(21,98),(22,99),(23,100),(24,101),(25,140),(26,141),(27,142),(28,143),(29,144),(30,133),(31,134),(32,135),(33,136),(34,137),(35,138),(36,139),(37,175),(38,176),(39,177),(40,178),(41,179),(42,180),(43,169),(44,170),(45,171),(46,172),(47,173),(48,174),(49,159),(50,160),(51,161),(52,162),(53,163),(54,164),(55,165),(56,166),(57,167),(58,168),(59,157),(60,158),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,181),(69,182),(70,183),(71,184),(72,185),(73,220),(74,221),(75,222),(76,223),(77,224),(78,225),(79,226),(80,227),(81,228),(82,217),(83,218),(84,219),(85,120),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(121,204),(122,193),(123,194),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(145,232),(146,233),(147,234),(148,235),(149,236),(150,237),(151,238),(152,239),(153,240),(154,229),(155,230),(156,231)], [(1,208),(2,209),(3,210),(4,211),(5,212),(6,213),(7,214),(8,215),(9,216),(10,205),(11,206),(12,207),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,97),(21,98),(22,99),(23,100),(24,101),(25,140),(26,141),(27,142),(28,143),(29,144),(30,133),(31,134),(32,135),(33,136),(34,137),(35,138),(36,139),(37,169),(38,170),(39,171),(40,172),(41,173),(42,174),(43,175),(44,176),(45,177),(46,178),(47,179),(48,180),(49,165),(50,166),(51,167),(52,168),(53,157),(54,158),(55,159),(56,160),(57,161),(58,162),(59,163),(60,164),(61,192),(62,181),(63,182),(64,183),(65,184),(66,185),(67,186),(68,187),(69,188),(70,189),(71,190),(72,191),(73,226),(74,227),(75,228),(76,217),(77,218),(78,219),(79,220),(80,221),(81,222),(82,223),(83,224),(84,225),(85,120),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(121,204),(122,193),(123,194),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(145,238),(146,239),(147,240),(148,229),(149,230),(150,231),(151,232),(152,233),(153,234),(154,235),(155,236),(156,237)], [(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)], [(1,188,214,63),(2,181,215,68),(3,186,216,61),(4,191,205,66),(5,184,206,71),(6,189,207,64),(7,182,208,69),(8,187,209,62),(9,192,210,67),(10,185,211,72),(11,190,212,65),(12,183,213,70),(13,219,108,84),(14,224,97,77),(15,217,98,82),(16,222,99,75),(17,227,100,80),(18,220,101,73),(19,225,102,78),(20,218,103,83),(21,223,104,76),(22,228,105,81),(23,221,106,74),(24,226,107,79),(25,156,134,231),(26,149,135,236),(27,154,136,229),(28,147,137,234),(29,152,138,239),(30,145,139,232),(31,150,140,237),(32,155,141,230),(33,148,142,235),(34,153,143,240),(35,146,144,233),(36,151,133,238),(37,197,175,132),(38,202,176,125),(39,195,177,130),(40,200,178,123),(41,193,179,128),(42,198,180,121),(43,203,169,126),(44,196,170,131),(45,201,171,124),(46,194,172,129),(47,199,173,122),(48,204,174,127),(49,109,159,92),(50,114,160,85),(51,119,161,90),(52,112,162,95),(53,117,163,88),(54,110,164,93),(55,115,165,86),(56,120,166,91),(57,113,167,96),(58,118,168,89),(59,111,157,94),(60,116,158,87)])

180 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G···4N5A5B5C5D6A···6G10A···10L10M···10T12A···12H15A15B15C15D20A···20P20Q···20X20Y···20BD30A···30AB60A···60AF
order12222234444444···455556···610···1010···1012···121515151520···2020···2020···2030···3060···60
size11112221111226···611112···21···12···22···222221···12···26···62···22···2

180 irreducible representations

dim111111111111222222222222
type++++++-++
imageC1C2C2C2C2C4C5C10C10C10C10C20S3Dic3D6D6C4○D4C5×S3C4○D12C5×Dic3S3×C10S3×C10C5×C4○D4C5×C4○D12
kernelC5×C23.26D6Dic3×C20C5×C4⋊Dic3C5×C6.D4C22×C60C2×C60C23.26D6C4×Dic3C4⋊Dic3C6.D4C22×C12C2×C12C22×C20C2×C20C2×C20C22×C10C30C22×C4C10C2×C4C2×C4C23C6C2
# reps1222184888432142144816841632

Matrix representation of C5×C23.26D6 in GL4(𝔽61) generated by

1000
0100
0090
0009
,
60000
50100
0010
00060
,
60000
06000
0010
0001
,
60000
06000
00600
00060
,
11000
01100
00210
00029
,
603900
0100
00029
00210
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[60,50,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,11,0,0,0,0,21,0,0,0,0,29],[60,0,0,0,39,1,0,0,0,0,0,21,0,0,29,0] >;

C5×C23.26D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{26}D_6
% in TeX

G:=Group("C5xC2^3.26D6");
// GroupNames label

G:=SmallGroup(480,805);
// by ID

G=gap.SmallGroup(480,805);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,1766,15686]);
// 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=e^5>;
// generators/relations

׿
×
𝔽