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## G = C5×C4⋊C4⋊S3order 480 = 25·3·5

### Direct product of C5 and C4⋊C4⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C4⋊C4⋊S3
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, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 276 in 120 conjugacy classes, 58 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4 [×3], C2×C4 [×3], C23, C10 [×3], C10, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×2], C20 [×6], C2×C10, C2×C10 [×3], C2×Dic3 [×3], C2×C12 [×3], C22×S3, C5×S3, C30 [×3], C422C2, C2×C20 [×3], C2×C20 [×3], C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C3×C4⋊C4, C5×Dic3 [×3], C60 [×3], S3×C10 [×3], C2×C30, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4, C5×C4⋊C4 [×2], C4⋊C4⋊S3, C10×Dic3 [×3], C2×C60 [×3], S3×C2×C10, C5×C422C2, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4 [×3], C15×C4⋊C4, C5×C4⋊C4⋊S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4 [×3], C2×C10 [×7], C22×S3, C5×S3, C422C2, C22×C10, C4○D12, D42S3, Q83S3, S3×C10 [×3], C5×C4○D4 [×3], C4⋊C4⋊S3, S3×C2×C10, C5×C422C2, C5×C4○D12, C5×D42S3, C5×Q83S3, C5×C4⋊C4⋊S3

Smallest permutation representation of C5×C4⋊C4⋊S3
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)
(1 103 58 126)(2 104 59 127)(3 105 60 128)(4 101 56 129)(5 102 57 130)(6 153 43 123)(7 154 44 124)(8 155 45 125)(9 151 41 121)(10 152 42 122)(11 141 48 171)(12 142 49 172)(13 143 50 173)(14 144 46 174)(15 145 47 175)(16 98 66 118)(17 99 67 119)(18 100 68 120)(19 96 69 116)(20 97 70 117)(21 180 225 160)(22 176 221 156)(23 177 222 157)(24 178 223 158)(25 179 224 159)(26 72 89 109)(27 73 90 110)(28 74 86 106)(29 75 87 107)(30 71 88 108)(31 78 51 149)(32 79 52 150)(33 80 53 146)(34 76 54 147)(35 77 55 148)(36 170 199 133)(37 166 200 134)(38 167 196 135)(39 168 197 131)(40 169 198 132)(61 82 91 112)(62 83 92 113)(63 84 93 114)(64 85 94 115)(65 81 95 111)(136 227 207 195)(137 228 208 191)(138 229 209 192)(139 230 210 193)(140 226 206 194)(161 236 189 216)(162 237 190 217)(163 238 186 218)(164 239 187 219)(165 240 188 220)(181 232 211 202)(182 233 212 203)(183 234 213 204)(184 235 214 205)(185 231 215 201)
(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 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)
(6 88)(7 89)(8 90)(9 86)(10 87)(11 199)(12 200)(13 196)(14 197)(15 198)(21 240)(22 236)(23 237)(24 238)(25 239)(26 44)(27 45)(28 41)(29 42)(30 43)(31 64)(32 65)(33 61)(34 62)(35 63)(36 48)(37 49)(38 50)(39 46)(40 47)(51 94)(52 95)(53 91)(54 92)(55 93)(71 114)(72 115)(73 111)(74 112)(75 113)(76 152)(77 153)(78 154)(79 155)(80 151)(81 110)(82 106)(83 107)(84 108)(85 109)(96 129)(97 130)(98 126)(99 127)(100 128)(101 116)(102 117)(103 118)(104 119)(105 120)(121 146)(122 147)(123 148)(124 149)(125 150)(131 213)(132 214)(133 215)(134 211)(135 212)(136 173)(137 174)(138 175)(139 171)(140 172)(141 210)(142 206)(143 207)(144 208)(145 209)(156 176)(157 177)(158 178)(159 179)(160 180)(161 189)(162 190)(163 186)(164 187)(165 188)(166 181)(167 182)(168 183)(169 184)(170 185)(191 204)(192 205)(193 201)(194 202)(195 203)(216 221)(217 222)(218 223)(219 224)(220 225)(226 232)(227 233)(228 234)(229 235)(230 231)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M 10N 10O 10P 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20AF 20AG 20AH 20AI 20AJ 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 10 10 10 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 12 2 2 2 4 4 6 6 6 6 12 1 1 1 1 2 2 2 1 ··· 1 12 12 12 12 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 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 C5 C10 C10 C10 C10 C10 S3 D6 C4○D4 C5×S3 C4○D12 S3×C10 C5×C4○D4 C5×C4○D12 D4⋊2S3 Q8⋊3S3 C5×D4⋊2S3 C5×Q8⋊3S3 kernel C5×C4⋊C4⋊S3 Dic3×C20 C5×Dic3⋊C4 C5×C4⋊Dic3 C5×D6⋊C4 C15×C4⋊C4 C4⋊C4⋊S3 C4×Dic3 Dic3⋊C4 C4⋊Dic3 D6⋊C4 C3×C4⋊C4 C5×C4⋊C4 C2×C20 C30 C4⋊C4 C10 C2×C4 C6 C2 C10 C10 C2 C2 # reps 1 1 1 1 3 1 4 4 4 4 12 4 1 3 6 4 4 12 24 16 1 1 4 4

Matrix representation of C5×C4⋊C4⋊S3 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 9 43 0 0 18 52 0 0 0 0 11 0 0 0 22 50
,
 50 0 0 0 0 50 0 0 0 0 2 59 0 0 32 59
,
 60 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 1 60 0 0 0 60 0 0 0 0 1 0 0 0 2 60
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[9,18,0,0,43,52,0,0,0,0,11,22,0,0,0,50],[50,0,0,0,0,50,0,0,0,0,2,32,0,0,59,59],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,60,60,0,0,0,0,1,2,0,0,0,60] >;

C5×C4⋊C4⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes C_4\rtimes S_3
% in TeX

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

G:=SmallGroup(480,777);
// by ID

G=gap.SmallGroup(480,777);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,288,926,891,436,15686]);
// 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,c*b*c^-1=b^-1,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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