direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4⋊C4⋊S3, D6⋊C4.4C10, C4⋊Dic3⋊7C10, (C2×C20).239D6, Dic3⋊C4⋊13C10, (Dic3×C20)⋊32C2, (C4×Dic3)⋊14C10, C15⋊23(C42⋊2C2), C30.207(C4○D4), (C2×C60).427C22, (C2×C30).418C23, C10.121(C4○D12), C10.52(Q8⋊3S3), C10.118(D4⋊2S3), (C10×Dic3).222C22, C4⋊C4⋊6(C5×S3), (C3×C4⋊C4)⋊9C10, (C5×C4⋊C4)⋊15S3, (C15×C4⋊C4)⋊27C2, C3⋊3(C5×C42⋊2C2), C6.34(C5×C4○D4), (C2×C4).31(S3×C10), (C5×D6⋊C4).11C2, (C5×C4⋊Dic3)⋊25C2, C2.16(C5×C4○D12), C22.53(S3×C2×C10), C2.7(C5×Q8⋊3S3), (C2×C12).58(C2×C10), (C5×Dic3⋊C4)⋊35C2, C2.14(C5×D4⋊2S3), (S3×C2×C10).69C22, (C22×S3).8(C2×C10), (C2×C6).39(C22×C10), (C2×C10).352(C22×S3), (C2×Dic3).30(C2×C10), SmallGroup(480,777)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C42⋊2C2, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C5×Dic3, C60, S3×C10, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C4⋊C4⋊S3, C10×Dic3, C2×C60, S3×C2×C10, C5×C42⋊2C2, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4, C15×C4⋊C4, C5×C4⋊C4⋊S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C42⋊2C2, C22×C10, C4○D12, D4⋊2S3, Q8⋊3S3, S3×C10, C5×C4○D4, C4⋊C4⋊S3, S3×C2×C10, C5×C42⋊2C2, C5×C4○D12, C5×D4⋊2S3, C5×Q8⋊3S3, 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 58 138)(2 104 59 139)(3 105 60 140)(4 101 56 136)(5 102 57 137)(6 82 63 107)(7 83 64 108)(8 84 65 109)(9 85 61 110)(10 81 62 106)(11 177 222 157)(12 178 223 158)(13 179 224 159)(14 180 225 160)(15 176 221 156)(16 151 41 150)(17 152 42 146)(18 153 43 147)(19 154 44 148)(20 155 45 149)(21 132 46 174)(22 133 47 175)(23 134 48 171)(24 135 49 172)(25 131 50 173)(26 98 78 118)(27 99 79 119)(28 100 80 120)(29 96 76 116)(30 97 77 117)(31 66 51 125)(32 67 52 121)(33 68 53 122)(34 69 54 123)(35 70 55 124)(36 170 187 145)(37 166 188 141)(38 167 189 142)(39 168 190 143)(40 169 186 144)(71 92 113 86)(72 93 114 87)(73 94 115 88)(74 95 111 89)(75 91 112 90)(126 229 185 204)(127 230 181 205)(128 226 182 201)(129 227 183 202)(130 228 184 203)(161 236 196 216)(162 237 197 217)(163 238 198 218)(164 239 199 219)(165 240 200 220)(191 206 233 212)(192 207 234 213)(193 208 235 214)(194 209 231 215)(195 210 232 211)
(1 198 78 158)(2 199 79 159)(3 200 80 160)(4 196 76 156)(5 197 77 157)(6 167 54 129)(7 168 55 130)(8 169 51 126)(9 170 52 127)(10 166 53 128)(11 137 217 97)(12 138 218 98)(13 139 219 99)(14 140 220 100)(15 136 216 96)(16 210 90 172)(17 206 86 173)(18 207 87 174)(19 208 88 175)(20 209 89 171)(21 147 234 114)(22 148 235 115)(23 149 231 111)(24 150 232 112)(25 146 233 113)(26 178 58 163)(27 179 59 164)(28 180 60 165)(29 176 56 161)(30 177 57 162)(31 185 65 144)(32 181 61 145)(33 182 62 141)(34 183 63 142)(35 184 64 143)(36 121 205 85)(37 122 201 81)(38 123 202 82)(39 124 203 83)(40 125 204 84)(41 211 91 135)(42 212 92 131)(43 213 93 132)(44 214 94 133)(45 215 95 134)(46 153 192 72)(47 154 193 73)(48 155 194 74)(49 151 195 75)(50 152 191 71)(66 229 109 186)(67 230 110 187)(68 226 106 188)(69 227 107 189)(70 228 108 190)(101 236 116 221)(102 237 117 222)(103 238 118 223)(104 239 119 224)(105 240 120 225)
(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)(11 237)(12 238)(13 239)(14 240)(15 236)(16 62)(17 63)(18 64)(19 65)(20 61)(21 190)(22 186)(23 187)(24 188)(25 189)(31 88)(32 89)(33 90)(34 86)(35 87)(36 48)(37 49)(38 50)(39 46)(40 47)(51 94)(52 95)(53 91)(54 92)(55 93)(66 154)(67 155)(68 151)(69 152)(70 153)(71 107)(72 108)(73 109)(74 110)(75 106)(81 112)(82 113)(83 114)(84 115)(85 111)(96 136)(97 137)(98 138)(99 139)(100 140)(101 116)(102 117)(103 118)(104 119)(105 120)(121 149)(122 150)(123 146)(124 147)(125 148)(126 175)(127 171)(128 172)(129 173)(130 174)(131 183)(132 184)(133 185)(134 181)(135 182)(141 211)(142 212)(143 213)(144 214)(145 215)(156 176)(157 177)(158 178)(159 179)(160 180)(161 196)(162 197)(163 198)(164 199)(165 200)(166 210)(167 206)(168 207)(169 208)(170 209)(191 202)(192 203)(193 204)(194 205)(195 201)(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), (1,103,58,138)(2,104,59,139)(3,105,60,140)(4,101,56,136)(5,102,57,137)(6,82,63,107)(7,83,64,108)(8,84,65,109)(9,85,61,110)(10,81,62,106)(11,177,222,157)(12,178,223,158)(13,179,224,159)(14,180,225,160)(15,176,221,156)(16,151,41,150)(17,152,42,146)(18,153,43,147)(19,154,44,148)(20,155,45,149)(21,132,46,174)(22,133,47,175)(23,134,48,171)(24,135,49,172)(25,131,50,173)(26,98,78,118)(27,99,79,119)(28,100,80,120)(29,96,76,116)(30,97,77,117)(31,66,51,125)(32,67,52,121)(33,68,53,122)(34,69,54,123)(35,70,55,124)(36,170,187,145)(37,166,188,141)(38,167,189,142)(39,168,190,143)(40,169,186,144)(71,92,113,86)(72,93,114,87)(73,94,115,88)(74,95,111,89)(75,91,112,90)(126,229,185,204)(127,230,181,205)(128,226,182,201)(129,227,183,202)(130,228,184,203)(161,236,196,216)(162,237,197,217)(163,238,198,218)(164,239,199,219)(165,240,200,220)(191,206,233,212)(192,207,234,213)(193,208,235,214)(194,209,231,215)(195,210,232,211), (1,198,78,158)(2,199,79,159)(3,200,80,160)(4,196,76,156)(5,197,77,157)(6,167,54,129)(7,168,55,130)(8,169,51,126)(9,170,52,127)(10,166,53,128)(11,137,217,97)(12,138,218,98)(13,139,219,99)(14,140,220,100)(15,136,216,96)(16,210,90,172)(17,206,86,173)(18,207,87,174)(19,208,88,175)(20,209,89,171)(21,147,234,114)(22,148,235,115)(23,149,231,111)(24,150,232,112)(25,146,233,113)(26,178,58,163)(27,179,59,164)(28,180,60,165)(29,176,56,161)(30,177,57,162)(31,185,65,144)(32,181,61,145)(33,182,62,141)(34,183,63,142)(35,184,64,143)(36,121,205,85)(37,122,201,81)(38,123,202,82)(39,124,203,83)(40,125,204,84)(41,211,91,135)(42,212,92,131)(43,213,93,132)(44,214,94,133)(45,215,95,134)(46,153,192,72)(47,154,193,73)(48,155,194,74)(49,151,195,75)(50,152,191,71)(66,229,109,186)(67,230,110,187)(68,226,106,188)(69,227,107,189)(70,228,108,190)(101,236,116,221)(102,237,117,222)(103,238,118,223)(104,239,119,224)(105,240,120,225), (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)(11,237)(12,238)(13,239)(14,240)(15,236)(16,62)(17,63)(18,64)(19,65)(20,61)(21,190)(22,186)(23,187)(24,188)(25,189)(31,88)(32,89)(33,90)(34,86)(35,87)(36,48)(37,49)(38,50)(39,46)(40,47)(51,94)(52,95)(53,91)(54,92)(55,93)(66,154)(67,155)(68,151)(69,152)(70,153)(71,107)(72,108)(73,109)(74,110)(75,106)(81,112)(82,113)(83,114)(84,115)(85,111)(96,136)(97,137)(98,138)(99,139)(100,140)(101,116)(102,117)(103,118)(104,119)(105,120)(121,149)(122,150)(123,146)(124,147)(125,148)(126,175)(127,171)(128,172)(129,173)(130,174)(131,183)(132,184)(133,185)(134,181)(135,182)(141,211)(142,212)(143,213)(144,214)(145,215)(156,176)(157,177)(158,178)(159,179)(160,180)(161,196)(162,197)(163,198)(164,199)(165,200)(166,210)(167,206)(168,207)(169,208)(170,209)(191,202)(192,203)(193,204)(194,205)(195,201)(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), (1,103,58,138)(2,104,59,139)(3,105,60,140)(4,101,56,136)(5,102,57,137)(6,82,63,107)(7,83,64,108)(8,84,65,109)(9,85,61,110)(10,81,62,106)(11,177,222,157)(12,178,223,158)(13,179,224,159)(14,180,225,160)(15,176,221,156)(16,151,41,150)(17,152,42,146)(18,153,43,147)(19,154,44,148)(20,155,45,149)(21,132,46,174)(22,133,47,175)(23,134,48,171)(24,135,49,172)(25,131,50,173)(26,98,78,118)(27,99,79,119)(28,100,80,120)(29,96,76,116)(30,97,77,117)(31,66,51,125)(32,67,52,121)(33,68,53,122)(34,69,54,123)(35,70,55,124)(36,170,187,145)(37,166,188,141)(38,167,189,142)(39,168,190,143)(40,169,186,144)(71,92,113,86)(72,93,114,87)(73,94,115,88)(74,95,111,89)(75,91,112,90)(126,229,185,204)(127,230,181,205)(128,226,182,201)(129,227,183,202)(130,228,184,203)(161,236,196,216)(162,237,197,217)(163,238,198,218)(164,239,199,219)(165,240,200,220)(191,206,233,212)(192,207,234,213)(193,208,235,214)(194,209,231,215)(195,210,232,211), (1,198,78,158)(2,199,79,159)(3,200,80,160)(4,196,76,156)(5,197,77,157)(6,167,54,129)(7,168,55,130)(8,169,51,126)(9,170,52,127)(10,166,53,128)(11,137,217,97)(12,138,218,98)(13,139,219,99)(14,140,220,100)(15,136,216,96)(16,210,90,172)(17,206,86,173)(18,207,87,174)(19,208,88,175)(20,209,89,171)(21,147,234,114)(22,148,235,115)(23,149,231,111)(24,150,232,112)(25,146,233,113)(26,178,58,163)(27,179,59,164)(28,180,60,165)(29,176,56,161)(30,177,57,162)(31,185,65,144)(32,181,61,145)(33,182,62,141)(34,183,63,142)(35,184,64,143)(36,121,205,85)(37,122,201,81)(38,123,202,82)(39,124,203,83)(40,125,204,84)(41,211,91,135)(42,212,92,131)(43,213,93,132)(44,214,94,133)(45,215,95,134)(46,153,192,72)(47,154,193,73)(48,155,194,74)(49,151,195,75)(50,152,191,71)(66,229,109,186)(67,230,110,187)(68,226,106,188)(69,227,107,189)(70,228,108,190)(101,236,116,221)(102,237,117,222)(103,238,118,223)(104,239,119,224)(105,240,120,225), (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)(11,237)(12,238)(13,239)(14,240)(15,236)(16,62)(17,63)(18,64)(19,65)(20,61)(21,190)(22,186)(23,187)(24,188)(25,189)(31,88)(32,89)(33,90)(34,86)(35,87)(36,48)(37,49)(38,50)(39,46)(40,47)(51,94)(52,95)(53,91)(54,92)(55,93)(66,154)(67,155)(68,151)(69,152)(70,153)(71,107)(72,108)(73,109)(74,110)(75,106)(81,112)(82,113)(83,114)(84,115)(85,111)(96,136)(97,137)(98,138)(99,139)(100,140)(101,116)(102,117)(103,118)(104,119)(105,120)(121,149)(122,150)(123,146)(124,147)(125,148)(126,175)(127,171)(128,172)(129,173)(130,174)(131,183)(132,184)(133,185)(134,181)(135,182)(141,211)(142,212)(143,213)(144,214)(145,215)(156,176)(157,177)(158,178)(159,179)(160,180)(161,196)(162,197)(163,198)(164,199)(165,200)(166,210)(167,206)(168,207)(169,208)(170,209)(191,202)(192,203)(193,204)(194,205)(195,201)(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)], [(1,103,58,138),(2,104,59,139),(3,105,60,140),(4,101,56,136),(5,102,57,137),(6,82,63,107),(7,83,64,108),(8,84,65,109),(9,85,61,110),(10,81,62,106),(11,177,222,157),(12,178,223,158),(13,179,224,159),(14,180,225,160),(15,176,221,156),(16,151,41,150),(17,152,42,146),(18,153,43,147),(19,154,44,148),(20,155,45,149),(21,132,46,174),(22,133,47,175),(23,134,48,171),(24,135,49,172),(25,131,50,173),(26,98,78,118),(27,99,79,119),(28,100,80,120),(29,96,76,116),(30,97,77,117),(31,66,51,125),(32,67,52,121),(33,68,53,122),(34,69,54,123),(35,70,55,124),(36,170,187,145),(37,166,188,141),(38,167,189,142),(39,168,190,143),(40,169,186,144),(71,92,113,86),(72,93,114,87),(73,94,115,88),(74,95,111,89),(75,91,112,90),(126,229,185,204),(127,230,181,205),(128,226,182,201),(129,227,183,202),(130,228,184,203),(161,236,196,216),(162,237,197,217),(163,238,198,218),(164,239,199,219),(165,240,200,220),(191,206,233,212),(192,207,234,213),(193,208,235,214),(194,209,231,215),(195,210,232,211)], [(1,198,78,158),(2,199,79,159),(3,200,80,160),(4,196,76,156),(5,197,77,157),(6,167,54,129),(7,168,55,130),(8,169,51,126),(9,170,52,127),(10,166,53,128),(11,137,217,97),(12,138,218,98),(13,139,219,99),(14,140,220,100),(15,136,216,96),(16,210,90,172),(17,206,86,173),(18,207,87,174),(19,208,88,175),(20,209,89,171),(21,147,234,114),(22,148,235,115),(23,149,231,111),(24,150,232,112),(25,146,233,113),(26,178,58,163),(27,179,59,164),(28,180,60,165),(29,176,56,161),(30,177,57,162),(31,185,65,144),(32,181,61,145),(33,182,62,141),(34,183,63,142),(35,184,64,143),(36,121,205,85),(37,122,201,81),(38,123,202,82),(39,124,203,83),(40,125,204,84),(41,211,91,135),(42,212,92,131),(43,213,93,132),(44,214,94,133),(45,215,95,134),(46,153,192,72),(47,154,193,73),(48,155,194,74),(49,151,195,75),(50,152,191,71),(66,229,109,186),(67,230,110,187),(68,226,106,188),(69,227,107,189),(70,228,108,190),(101,236,116,221),(102,237,117,222),(103,238,118,223),(104,239,119,224),(105,240,120,225)], [(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),(11,237),(12,238),(13,239),(14,240),(15,236),(16,62),(17,63),(18,64),(19,65),(20,61),(21,190),(22,186),(23,187),(24,188),(25,189),(31,88),(32,89),(33,90),(34,86),(35,87),(36,48),(37,49),(38,50),(39,46),(40,47),(51,94),(52,95),(53,91),(54,92),(55,93),(66,154),(67,155),(68,151),(69,152),(70,153),(71,107),(72,108),(73,109),(74,110),(75,106),(81,112),(82,113),(83,114),(84,115),(85,111),(96,136),(97,137),(98,138),(99,139),(100,140),(101,116),(102,117),(103,118),(104,119),(105,120),(121,149),(122,150),(123,146),(124,147),(125,148),(126,175),(127,171),(128,172),(129,173),(130,174),(131,183),(132,184),(133,185),(134,181),(135,182),(141,211),(142,212),(143,213),(144,214),(145,215),(156,176),(157,177),(158,178),(159,179),(160,180),(161,196),(162,197),(163,198),(164,199),(165,200),(166,210),(167,206),(168,207),(169,208),(170,209),(191,202),(192,203),(193,204),(194,205),(195,201),(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