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## G = C5×C42⋊7S3order 480 = 25·3·5

### Direct product of C5 and C42⋊7S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C42⋊7S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C5×D6⋊C4 — C5×C42⋊7S3
 Lower central C3 — C2×C6 — C5×C42⋊7S3
 Upper central C1 — C2×C10 — C4×C20

Generators and relations for C5×C427S3
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, ede=d-1 >

Subgroups: 404 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C5×S3 [×2], C30, C30 [×2], C4.4D4, C2×C20, C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], D6⋊C4 [×4], C4×C12, C2×Dic6, C2×D12, C5×Dic3 [×2], C60 [×2], C60 [×2], S3×C10 [×6], C2×C30, C4×C20, C5×C22⋊C4 [×4], D4×C10, Q8×C10, C427S3, C5×Dic6 [×2], C5×D12 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], S3×C2×C10 [×2], C5×C4.4D4, C5×D6⋊C4 [×4], C4×C60, C10×Dic6, C10×D12, C5×C427S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C4.4D4, C5×D4 [×2], C22×C10, C2×D12, C4○D12 [×2], S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C427S3, C5×D12 [×2], S3×C2×C10, C5×C4.4D4, C10×D12, C5×C4○D12 [×2], C5×C427S3

Smallest permutation representation of C5×C427S3
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 66 158)(2 187 67 159)(3 188 68 160)(4 189 69 156)(5 190 70 157)(6 183 63 174)(7 184 64 175)(8 185 65 171)(9 181 61 172)(10 182 62 173)(11 125 231 111)(12 121 232 112)(13 122 233 113)(14 123 234 114)(15 124 235 115)(16 178 58 163)(17 179 59 164)(18 180 60 165)(19 176 56 161)(20 177 57 162)(21 128 220 100)(22 129 216 96)(23 130 217 97)(24 126 218 98)(25 127 219 99)(26 169 51 138)(27 170 52 139)(28 166 53 140)(29 167 54 136)(30 168 55 137)(31 209 89 132)(32 210 90 133)(33 206 86 134)(34 207 87 135)(35 208 88 131)(36 150 193 73)(37 146 194 74)(38 147 195 75)(39 148 191 71)(40 149 192 72)(41 211 91 142)(42 212 92 143)(43 213 93 144)(44 214 94 145)(45 215 95 141)(46 153 204 84)(47 154 205 85)(48 155 201 81)(49 151 202 82)(50 152 203 83)(76 227 107 196)(77 228 108 197)(78 229 109 198)(79 230 110 199)(80 226 106 200)(101 236 116 221)(102 237 117 222)(103 238 118 223)(104 239 119 224)(105 240 120 225)
(1 118 58 98)(2 119 59 99)(3 120 60 100)(4 116 56 96)(5 117 57 97)(6 84 43 114)(7 85 44 115)(8 81 45 111)(9 82 41 112)(10 83 42 113)(11 185 48 215)(12 181 49 211)(13 182 50 212)(14 183 46 213)(15 184 47 214)(16 126 66 103)(17 127 67 104)(18 128 68 105)(19 129 69 101)(20 130 70 102)(21 188 225 165)(22 189 221 161)(23 190 222 162)(24 186 223 163)(25 187 224 164)(26 149 89 78)(27 150 90 79)(28 146 86 80)(29 147 87 76)(30 148 88 77)(31 109 51 72)(32 110 52 73)(33 106 53 74)(34 107 54 75)(35 108 55 71)(36 210 199 139)(37 206 200 140)(38 207 196 136)(39 208 197 137)(40 209 198 138)(61 151 91 121)(62 152 92 122)(63 153 93 123)(64 154 94 124)(65 155 95 125)(131 228 168 191)(132 229 169 192)(133 230 170 193)(134 226 166 194)(135 227 167 195)(141 231 171 201)(142 232 172 202)(143 233 173 203)(144 234 174 204)(145 235 175 205)(156 236 176 216)(157 237 177 217)(158 238 178 218)(159 239 179 219)(160 240 180 220)
(1 8 90)(2 9 86)(3 10 87)(4 6 88)(5 7 89)(11 193 24)(12 194 25)(13 195 21)(14 191 22)(15 192 23)(16 95 52)(17 91 53)(18 92 54)(19 93 55)(20 94 51)(26 57 44)(27 58 45)(28 59 41)(29 60 42)(30 56 43)(31 70 64)(32 66 65)(33 67 61)(34 68 62)(35 69 63)(36 218 231)(37 219 232)(38 220 233)(39 216 234)(40 217 235)(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)(76 120 83)(77 116 84)(78 117 85)(79 118 81)(80 119 82)(96 114 148)(97 115 149)(98 111 150)(99 112 146)(100 113 147)(101 153 108)(102 154 109)(103 155 110)(104 151 106)(105 152 107)(131 189 183)(132 190 184)(133 186 185)(134 187 181)(135 188 182)(136 180 143)(137 176 144)(138 177 145)(139 178 141)(140 179 142)(156 174 208)(157 175 209)(158 171 210)(159 172 206)(160 173 207)(161 213 168)(162 214 169)(163 215 170)(164 211 166)(165 212 167)(196 240 203)(197 236 204)(198 237 205)(199 238 201)(200 239 202)
(1 218)(2 219)(3 220)(4 216)(5 217)(6 39)(7 40)(8 36)(9 37)(10 38)(11 32)(12 33)(13 34)(14 35)(15 31)(16 223)(17 224)(18 225)(19 221)(20 222)(21 68)(22 69)(23 70)(24 66)(25 67)(26 205)(27 201)(28 202)(29 203)(30 204)(41 200)(42 196)(43 197)(44 198)(45 199)(46 55)(47 51)(48 52)(49 53)(50 54)(56 236)(57 237)(58 238)(59 239)(60 240)(61 194)(62 195)(63 191)(64 192)(65 193)(71 144)(72 145)(73 141)(74 142)(75 143)(76 182)(77 183)(78 184)(79 185)(80 181)(81 133)(82 134)(83 135)(84 131)(85 132)(86 232)(87 233)(88 234)(89 235)(90 231)(91 226)(92 227)(93 228)(94 229)(95 230)(96 161)(97 162)(98 163)(99 164)(100 165)(101 156)(102 157)(103 158)(104 159)(105 160)(106 172)(107 173)(108 174)(109 175)(110 171)(111 170)(112 166)(113 167)(114 168)(115 169)(116 189)(117 190)(118 186)(119 187)(120 188)(121 140)(122 136)(123 137)(124 138)(125 139)(126 178)(127 179)(128 180)(129 176)(130 177)(146 211)(147 212)(148 213)(149 214)(150 215)(151 206)(152 207)(153 208)(154 209)(155 210)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10T 12A ··· 12L 15A 15B 15C 15D 20A ··· 20X 20Y ··· 20AF 30A ··· 30L 60A ··· 60AV order 1 2 2 2 2 2 3 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 30 ··· 30 60 ··· 60 size 1 1 1 1 12 12 2 2 ··· 2 12 12 1 1 1 1 2 2 2 1 ··· 1 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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D6 C4○D4 D12 C5×S3 C5×D4 C4○D12 S3×C10 C5×C4○D4 C5×D12 C5×C4○D12 kernel C5×C42⋊7S3 C5×D6⋊C4 C4×C60 C10×Dic6 C10×D12 C42⋊7S3 D6⋊C4 C4×C12 C2×Dic6 C2×D12 C4×C20 C60 C2×C20 C30 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 1 1 1 4 16 4 4 4 1 2 3 4 4 4 8 8 12 16 16 32

Matrix representation of C5×C427S3 in GL4(𝔽61) generated by

 20 0 0 0 0 20 0 0 0 0 9 0 0 0 0 9
,
 38 15 0 0 46 23 0 0 0 0 11 0 0 0 0 11
,
 23 46 0 0 15 38 0 0 0 0 32 18 0 0 55 29
,
 0 1 0 0 60 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 60 60 0 0 0 0 14 22 0 0 5 47
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,9,0,0,0,0,9],[38,46,0,0,15,23,0,0,0,0,11,0,0,0,0,11],[23,15,0,0,46,38,0,0,0,0,32,55,0,0,18,29],[0,60,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,0,60,0,0,0,0,14,5,0,0,22,47] >;

C5×C427S3 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_7S_3
% in TeX

G:=Group("C5xC4^2:7S3");
// GroupNames label

G:=SmallGroup(480,754);
// by ID

G=gap.SmallGroup(480,754);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,288,926,436,15686]);
// 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,e*d*e=d^-1>;
// generators/relations

׿
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𝔽