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## G = C5×C4⋊C4⋊7S3order 480 = 25·3·5

### Direct product of C5 and C4⋊C4⋊7S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C4⋊C47S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 308 in 152 conjugacy classes, 82 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, C20 [×2], C20 [×6], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C42⋊C2, C2×C20, C2×C20 [×2], C2×C20 [×7], C22×C10, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C5×Dic3 [×2], C5×Dic3 [×2], C60 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4, C22×C20, C4⋊C47S3, S3×C20 [×4], C10×Dic3, C10×Dic3 [×2], C2×C60, C2×C60 [×2], S3×C2×C10, C5×C42⋊C2, Dic3×C20 [×2], C5×C4⋊Dic3, C5×D6⋊C4 [×2], C15×C4⋊C4, S3×C2×C20, C5×C4⋊C47S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], C22×C4, C4○D4 [×2], C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C42⋊C2, C2×C20 [×6], C22×C10, S3×C2×C4, D42S3, Q83S3, S3×C10 [×3], C22×C20, C5×C4○D4 [×2], C4⋊C47S3, S3×C20 [×2], S3×C2×C10, C5×C42⋊C2, S3×C2×C20, C5×D42S3, C5×Q83S3, C5×C4⋊C47S3

Smallest permutation representation of C5×C4⋊C47S3
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)
(1 163 58 198)(2 164 59 199)(3 165 60 200)(4 161 56 196)(5 162 57 197)(6 142 63 167)(7 143 64 168)(8 144 65 169)(9 145 61 170)(10 141 62 166)(11 97 222 117)(12 98 223 118)(13 99 224 119)(14 100 225 120)(15 96 221 116)(16 211 41 210)(17 212 42 206)(18 213 43 207)(19 214 44 208)(20 215 45 209)(21 114 46 72)(22 115 47 73)(23 111 48 74)(24 112 49 75)(25 113 50 71)(26 158 78 178)(27 159 79 179)(28 160 80 180)(29 156 76 176)(30 157 77 177)(31 126 51 185)(32 127 52 181)(33 128 53 182)(34 129 54 183)(35 130 55 184)(36 85 187 110)(37 81 188 106)(38 82 189 107)(39 83 190 108)(40 84 186 109)(66 204 125 229)(67 205 121 230)(68 201 122 226)(69 202 123 227)(70 203 124 228)(86 131 92 173)(87 132 93 174)(88 133 94 175)(89 134 95 171)(90 135 91 172)(101 216 136 236)(102 217 137 237)(103 218 138 238)(104 219 139 239)(105 220 140 240)(146 191 152 233)(147 192 153 234)(148 193 154 235)(149 194 155 231)(150 195 151 232)
(1 138 78 98)(2 139 79 99)(3 140 80 100)(4 136 76 96)(5 137 77 97)(6 107 54 69)(7 108 55 70)(8 109 51 66)(9 110 52 67)(10 106 53 68)(11 162 217 177)(12 163 218 178)(13 164 219 179)(14 165 220 180)(15 161 216 176)(16 150 90 112)(17 146 86 113)(18 147 87 114)(19 148 88 115)(20 149 89 111)(21 213 234 132)(22 214 235 133)(23 215 231 134)(24 211 232 135)(25 212 233 131)(26 118 58 103)(27 119 59 104)(28 120 60 105)(29 116 56 101)(30 117 57 102)(31 125 65 84)(32 121 61 85)(33 122 62 81)(34 123 63 82)(35 124 64 83)(36 127 205 170)(37 128 201 166)(38 129 202 167)(39 130 203 168)(40 126 204 169)(41 151 91 75)(42 152 92 71)(43 153 93 72)(44 154 94 73)(45 155 95 74)(46 207 192 174)(47 208 193 175)(48 209 194 171)(49 210 195 172)(50 206 191 173)(141 188 182 226)(142 189 183 227)(143 190 184 228)(144 186 185 229)(145 187 181 230)(156 221 196 236)(157 222 197 237)(158 223 198 238)(159 224 199 239)(160 225 200 240)
(1 20 61)(2 16 62)(3 17 63)(4 18 64)(5 19 65)(6 60 42)(7 56 43)(8 57 44)(9 58 45)(10 59 41)(11 22 204)(12 23 205)(13 24 201)(14 25 202)(15 21 203)(26 95 52)(27 91 53)(28 92 54)(29 93 55)(30 94 51)(31 77 88)(32 78 89)(33 79 90)(34 80 86)(35 76 87)(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)(66 117 73)(67 118 74)(68 119 75)(69 120 71)(70 116 72)(81 139 150)(82 140 146)(83 136 147)(84 137 148)(85 138 149)(96 114 124)(97 115 125)(98 111 121)(99 112 122)(100 113 123)(101 153 108)(102 154 109)(103 155 110)(104 151 106)(105 152 107)(126 177 133)(127 178 134)(128 179 135)(129 180 131)(130 176 132)(141 199 210)(142 200 206)(143 196 207)(144 197 208)(145 198 209)(156 174 184)(157 175 185)(158 171 181)(159 172 182)(160 173 183)(161 213 168)(162 214 169)(163 215 170)(164 211 166)(165 212 167)(186 237 193)(187 238 194)(188 239 195)(189 240 191)(190 236 192)
(6 42)(7 43)(8 44)(9 45)(10 41)(11 222)(12 223)(13 224)(14 225)(15 221)(16 62)(17 63)(18 64)(19 65)(20 61)(21 228)(22 229)(23 230)(24 226)(25 227)(31 88)(32 89)(33 90)(34 86)(35 87)(36 194)(37 195)(38 191)(39 192)(40 193)(46 203)(47 204)(48 205)(49 201)(50 202)(51 94)(52 95)(53 91)(54 92)(55 93)(66 115)(67 111)(68 112)(69 113)(70 114)(71 123)(72 124)(73 125)(74 121)(75 122)(81 151)(82 152)(83 153)(84 154)(85 155)(96 116)(97 117)(98 118)(99 119)(100 120)(101 136)(102 137)(103 138)(104 139)(105 140)(106 150)(107 146)(108 147)(109 148)(110 149)(126 133)(127 134)(128 135)(129 131)(130 132)(141 210)(142 206)(143 207)(144 208)(145 209)(166 211)(167 212)(168 213)(169 214)(170 215)(171 181)(172 182)(173 183)(174 184)(175 185)(186 235)(187 231)(188 232)(189 233)(190 234)(216 236)(217 237)(218 238)(219 239)(220 240)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10T 12A ··· 12F 15A 15B 15C 15D 20A ··· 20X 20Y ··· 20AN 20AO ··· 20BD 30A ··· 30L 60A ··· 60X 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 2 ··· 2 3 3 3 3 6 6 6 6 1 1 1 1 2 2 2 1 ··· 1 6 ··· 6 4 ··· 4 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4

150 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 4 4 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 S3 D6 C4○D4 C4×S3 C5×S3 S3×C10 C5×C4○D4 S3×C20 D4⋊2S3 Q8⋊3S3 C5×D4⋊2S3 C5×Q8⋊3S3 kernel C5×C4⋊C4⋊7S3 Dic3×C20 C5×C4⋊Dic3 C5×D6⋊C4 C15×C4⋊C4 S3×C2×C20 S3×C20 C4⋊C4⋊7S3 C4×Dic3 C4⋊Dic3 D6⋊C4 C3×C4⋊C4 S3×C2×C4 C4×S3 C5×C4⋊C4 C2×C20 C30 C20 C4⋊C4 C2×C4 C6 C4 C10 C10 C2 C2 # reps 1 2 1 2 1 1 8 4 8 4 8 4 4 32 1 3 4 4 4 12 16 16 1 1 4 4

Matrix representation of C5×C4⋊C47S3 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 58 0 0 0 0 58
,
 60 0 0 0 0 60 0 0 0 0 11 0 0 0 0 50
,
 11 0 0 0 0 11 0 0 0 0 0 1 0 0 60 0
,
 60 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 60
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,50],[11,0,0,0,0,11,0,0,0,0,0,60,0,0,1,0],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;

C5×C4⋊C47S3 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes C_4\rtimes_7S_3
% in TeX

G:=Group("C5xC4:C4:7S3");
// GroupNames label

G:=SmallGroup(480,771);
// by ID

G=gap.SmallGroup(480,771);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,1766,891,226,15686]);
// 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,c*b*c^-1=b^-1,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

׿
×
𝔽