direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4⋊C4⋊7S3, (C4×S3)⋊2C20, (S3×C20)⋊10C4, C4.14(S3×C20), D6⋊C4.3C10, D6.4(C2×C20), C20.116(C4×S3), C60.181(C2×C4), C12.11(C2×C20), C4⋊Dic3⋊12C10, (C2×C20).276D6, (Dic3×C20)⋊31C2, (C4×Dic3)⋊13C10, C6.10(C22×C20), Dic3.9(C2×C20), C15⋊29(C42⋊C2), C30.250(C4○D4), (C2×C30).412C23, (C2×C60).417C22, C30.201(C22×C4), C10.48(Q8⋊3S3), C10.116(D4⋊2S3), (C10×Dic3).219C22, C4⋊C4⋊7(C5×S3), (C3×C4⋊C4)⋊3C10, (C5×C4⋊C4)⋊16S3, (S3×C2×C4).2C10, C2.12(S3×C2×C20), (C15×C4⋊C4)⋊21C2, (S3×C2×C20).13C2, C10.137(S3×C2×C4), C3⋊3(C5×C42⋊C2), C6.25(C5×C4○D4), (C2×C4).29(S3×C10), (C5×D6⋊C4).14C2, (C5×C4⋊Dic3)⋊30C2, C2.4(C5×D4⋊2S3), C22.17(S3×C2×C10), C2.1(C5×Q8⋊3S3), (S3×C10).40(C2×C4), (C2×C12).22(C2×C10), (S3×C2×C10).110C22, (C2×C6).33(C22×C10), (C5×Dic3).51(C2×C4), (C22×S3).19(C2×C10), (C2×C10).346(C22×S3), (C2×Dic3).48(C2×C10), SmallGroup(480,771)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊C4⋊7S3
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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C42⋊C2, C2×C20, C2×C20, C2×C20, C22×C10, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C4⋊C4⋊7S3, S3×C20, C10×Dic3, C10×Dic3, C2×C60, C2×C60, S3×C2×C10, C5×C42⋊C2, Dic3×C20, C5×C4⋊Dic3, C5×D6⋊C4, C15×C4⋊C4, S3×C2×C20, C5×C4⋊C4⋊7S3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C42⋊C2, C2×C20, C22×C10, S3×C2×C4, D4⋊2S3, Q8⋊3S3, S3×C10, C22×C20, C5×C4○D4, C4⋊C4⋊7S3, S3×C20, S3×C2×C10, C5×C42⋊C2, S3×C2×C20, C5×D4⋊2S3, C5×Q8⋊3S3, C5×C4⋊C4⋊7S3
►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 58 186)(2 164 59 187)(3 165 60 188)(4 161 56 189)(5 162 57 190)(6 133 90 170)(7 134 86 166)(8 135 87 167)(9 131 88 168)(10 132 89 169)(11 111 48 98)(12 112 49 99)(13 113 50 100)(14 114 46 96)(15 115 47 97)(16 152 53 206)(17 153 54 207)(18 154 55 208)(19 155 51 209)(20 151 52 210)(21 75 213 107)(22 71 214 108)(23 72 215 109)(24 73 211 110)(25 74 212 106)(26 141 66 178)(27 142 67 179)(28 143 68 180)(29 144 69 176)(30 145 70 177)(31 196 43 183)(32 197 44 184)(33 198 45 185)(34 199 41 181)(35 200 42 182)(36 81 223 118)(37 82 224 119)(38 83 225 120)(39 84 221 116)(40 85 222 117)(61 159 79 172)(62 160 80 173)(63 156 76 174)(64 157 77 175)(65 158 78 171)(91 193 150 230)(92 194 146 226)(93 195 147 227)(94 191 148 228)(95 192 149 229)(101 204 129 236)(102 205 130 237)(103 201 126 238)(104 202 127 239)(105 203 128 240)(121 219 139 232)(122 220 140 233)(123 216 136 234)(124 217 137 235)(125 218 138 231)
(1 126 66 81)(2 127 67 82)(3 128 68 83)(4 129 69 84)(5 130 70 85)(6 110 52 91)(7 106 53 92)(8 107 54 93)(9 108 55 94)(10 109 51 95)(11 198 231 158)(12 199 232 159)(13 200 233 160)(14 196 234 156)(15 197 235 157)(16 146 86 74)(17 147 87 75)(18 148 88 71)(19 149 89 72)(20 150 90 73)(21 153 195 167)(22 154 191 168)(23 155 192 169)(24 151 193 170)(25 152 194 166)(26 118 58 103)(27 119 59 104)(28 120 60 105)(29 116 56 101)(30 117 57 102)(31 123 63 114)(32 124 64 115)(33 125 65 111)(34 121 61 112)(35 122 62 113)(36 163 201 178)(37 164 202 179)(38 165 203 180)(39 161 204 176)(40 162 205 177)(41 139 79 99)(42 140 80 100)(43 136 76 96)(44 137 77 97)(45 138 78 98)(46 183 216 174)(47 184 217 175)(48 185 218 171)(49 181 219 172)(50 182 220 173)(131 214 208 228)(132 215 209 229)(133 211 210 230)(134 212 206 226)(135 213 207 227)(141 223 186 238)(142 224 187 239)(143 225 188 240)(144 221 189 236)(145 222 190 237)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 193 36)(12 194 37)(13 195 38)(14 191 39)(15 192 40)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 203 233)(22 204 234)(23 205 235)(24 201 231)(25 202 232)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(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)(81 111 150)(82 112 146)(83 113 147)(84 114 148)(85 115 149)(91 118 98)(92 119 99)(93 120 100)(94 116 96)(95 117 97)(101 136 108)(102 137 109)(103 138 110)(104 139 106)(105 140 107)(131 189 183)(132 190 184)(133 186 185)(134 187 181)(135 188 182)(141 171 210)(142 172 206)(143 173 207)(144 174 208)(145 175 209)(151 178 158)(152 179 159)(153 180 160)(154 176 156)(155 177 157)(161 196 168)(162 197 169)(163 198 170)(164 199 166)(165 200 167)(211 238 218)(212 239 219)(213 240 220)(214 236 216)(215 237 217)
(6 45)(7 41)(8 42)(9 43)(10 44)(11 230)(12 226)(13 227)(14 228)(15 229)(16 61)(17 62)(18 63)(19 64)(20 65)(21 220)(22 216)(23 217)(24 218)(25 219)(31 88)(32 89)(33 90)(34 86)(35 87)(36 223)(37 224)(38 225)(39 221)(40 222)(46 191)(47 192)(48 193)(49 194)(50 195)(51 77)(52 78)(53 79)(54 80)(55 76)(71 136)(72 137)(73 138)(74 139)(75 140)(81 118)(82 119)(83 120)(84 116)(85 117)(91 111)(92 112)(93 113)(94 114)(95 115)(96 148)(97 149)(98 150)(99 146)(100 147)(101 129)(102 130)(103 126)(104 127)(105 128)(106 121)(107 122)(108 123)(109 124)(110 125)(131 183)(132 184)(133 185)(134 181)(135 182)(151 158)(152 159)(153 160)(154 156)(155 157)(166 199)(167 200)(168 196)(169 197)(170 198)(171 210)(172 206)(173 207)(174 208)(175 209)(201 238)(202 239)(203 240)(204 236)(205 237)(211 231)(212 232)(213 233)(214 234)(215 235)
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), (1,163,58,186)(2,164,59,187)(3,165,60,188)(4,161,56,189)(5,162,57,190)(6,133,90,170)(7,134,86,166)(8,135,87,167)(9,131,88,168)(10,132,89,169)(11,111,48,98)(12,112,49,99)(13,113,50,100)(14,114,46,96)(15,115,47,97)(16,152,53,206)(17,153,54,207)(18,154,55,208)(19,155,51,209)(20,151,52,210)(21,75,213,107)(22,71,214,108)(23,72,215,109)(24,73,211,110)(25,74,212,106)(26,141,66,178)(27,142,67,179)(28,143,68,180)(29,144,69,176)(30,145,70,177)(31,196,43,183)(32,197,44,184)(33,198,45,185)(34,199,41,181)(35,200,42,182)(36,81,223,118)(37,82,224,119)(38,83,225,120)(39,84,221,116)(40,85,222,117)(61,159,79,172)(62,160,80,173)(63,156,76,174)(64,157,77,175)(65,158,78,171)(91,193,150,230)(92,194,146,226)(93,195,147,227)(94,191,148,228)(95,192,149,229)(101,204,129,236)(102,205,130,237)(103,201,126,238)(104,202,127,239)(105,203,128,240)(121,219,139,232)(122,220,140,233)(123,216,136,234)(124,217,137,235)(125,218,138,231), (1,126,66,81)(2,127,67,82)(3,128,68,83)(4,129,69,84)(5,130,70,85)(6,110,52,91)(7,106,53,92)(8,107,54,93)(9,108,55,94)(10,109,51,95)(11,198,231,158)(12,199,232,159)(13,200,233,160)(14,196,234,156)(15,197,235,157)(16,146,86,74)(17,147,87,75)(18,148,88,71)(19,149,89,72)(20,150,90,73)(21,153,195,167)(22,154,191,168)(23,155,192,169)(24,151,193,170)(25,152,194,166)(26,118,58,103)(27,119,59,104)(28,120,60,105)(29,116,56,101)(30,117,57,102)(31,123,63,114)(32,124,64,115)(33,125,65,111)(34,121,61,112)(35,122,62,113)(36,163,201,178)(37,164,202,179)(38,165,203,180)(39,161,204,176)(40,162,205,177)(41,139,79,99)(42,140,80,100)(43,136,76,96)(44,137,77,97)(45,138,78,98)(46,183,216,174)(47,184,217,175)(48,185,218,171)(49,181,219,172)(50,182,220,173)(131,214,208,228)(132,215,209,229)(133,211,210,230)(134,212,206,226)(135,213,207,227)(141,223,186,238)(142,224,187,239)(143,225,188,240)(144,221,189,236)(145,222,190,237), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,193,36)(12,194,37)(13,195,38)(14,191,39)(15,192,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,203,233)(22,204,234)(23,205,235)(24,201,231)(25,202,232)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(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)(81,111,150)(82,112,146)(83,113,147)(84,114,148)(85,115,149)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97)(101,136,108)(102,137,109)(103,138,110)(104,139,106)(105,140,107)(131,189,183)(132,190,184)(133,186,185)(134,187,181)(135,188,182)(141,171,210)(142,172,206)(143,173,207)(144,174,208)(145,175,209)(151,178,158)(152,179,159)(153,180,160)(154,176,156)(155,177,157)(161,196,168)(162,197,169)(163,198,170)(164,199,166)(165,200,167)(211,238,218)(212,239,219)(213,240,220)(214,236,216)(215,237,217), (6,45)(7,41)(8,42)(9,43)(10,44)(11,230)(12,226)(13,227)(14,228)(15,229)(16,61)(17,62)(18,63)(19,64)(20,65)(21,220)(22,216)(23,217)(24,218)(25,219)(31,88)(32,89)(33,90)(34,86)(35,87)(36,223)(37,224)(38,225)(39,221)(40,222)(46,191)(47,192)(48,193)(49,194)(50,195)(51,77)(52,78)(53,79)(54,80)(55,76)(71,136)(72,137)(73,138)(74,139)(75,140)(81,118)(82,119)(83,120)(84,116)(85,117)(91,111)(92,112)(93,113)(94,114)(95,115)(96,148)(97,149)(98,150)(99,146)(100,147)(101,129)(102,130)(103,126)(104,127)(105,128)(106,121)(107,122)(108,123)(109,124)(110,125)(131,183)(132,184)(133,185)(134,181)(135,182)(151,158)(152,159)(153,160)(154,156)(155,157)(166,199)(167,200)(168,196)(169,197)(170,198)(171,210)(172,206)(173,207)(174,208)(175,209)(201,238)(202,239)(203,240)(204,236)(205,237)(211,231)(212,232)(213,233)(214,234)(215,235)>;
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), (1,163,58,186)(2,164,59,187)(3,165,60,188)(4,161,56,189)(5,162,57,190)(6,133,90,170)(7,134,86,166)(8,135,87,167)(9,131,88,168)(10,132,89,169)(11,111,48,98)(12,112,49,99)(13,113,50,100)(14,114,46,96)(15,115,47,97)(16,152,53,206)(17,153,54,207)(18,154,55,208)(19,155,51,209)(20,151,52,210)(21,75,213,107)(22,71,214,108)(23,72,215,109)(24,73,211,110)(25,74,212,106)(26,141,66,178)(27,142,67,179)(28,143,68,180)(29,144,69,176)(30,145,70,177)(31,196,43,183)(32,197,44,184)(33,198,45,185)(34,199,41,181)(35,200,42,182)(36,81,223,118)(37,82,224,119)(38,83,225,120)(39,84,221,116)(40,85,222,117)(61,159,79,172)(62,160,80,173)(63,156,76,174)(64,157,77,175)(65,158,78,171)(91,193,150,230)(92,194,146,226)(93,195,147,227)(94,191,148,228)(95,192,149,229)(101,204,129,236)(102,205,130,237)(103,201,126,238)(104,202,127,239)(105,203,128,240)(121,219,139,232)(122,220,140,233)(123,216,136,234)(124,217,137,235)(125,218,138,231), (1,126,66,81)(2,127,67,82)(3,128,68,83)(4,129,69,84)(5,130,70,85)(6,110,52,91)(7,106,53,92)(8,107,54,93)(9,108,55,94)(10,109,51,95)(11,198,231,158)(12,199,232,159)(13,200,233,160)(14,196,234,156)(15,197,235,157)(16,146,86,74)(17,147,87,75)(18,148,88,71)(19,149,89,72)(20,150,90,73)(21,153,195,167)(22,154,191,168)(23,155,192,169)(24,151,193,170)(25,152,194,166)(26,118,58,103)(27,119,59,104)(28,120,60,105)(29,116,56,101)(30,117,57,102)(31,123,63,114)(32,124,64,115)(33,125,65,111)(34,121,61,112)(35,122,62,113)(36,163,201,178)(37,164,202,179)(38,165,203,180)(39,161,204,176)(40,162,205,177)(41,139,79,99)(42,140,80,100)(43,136,76,96)(44,137,77,97)(45,138,78,98)(46,183,216,174)(47,184,217,175)(48,185,218,171)(49,181,219,172)(50,182,220,173)(131,214,208,228)(132,215,209,229)(133,211,210,230)(134,212,206,226)(135,213,207,227)(141,223,186,238)(142,224,187,239)(143,225,188,240)(144,221,189,236)(145,222,190,237), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,193,36)(12,194,37)(13,195,38)(14,191,39)(15,192,40)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,203,233)(22,204,234)(23,205,235)(24,201,231)(25,202,232)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(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)(81,111,150)(82,112,146)(83,113,147)(84,114,148)(85,115,149)(91,118,98)(92,119,99)(93,120,100)(94,116,96)(95,117,97)(101,136,108)(102,137,109)(103,138,110)(104,139,106)(105,140,107)(131,189,183)(132,190,184)(133,186,185)(134,187,181)(135,188,182)(141,171,210)(142,172,206)(143,173,207)(144,174,208)(145,175,209)(151,178,158)(152,179,159)(153,180,160)(154,176,156)(155,177,157)(161,196,168)(162,197,169)(163,198,170)(164,199,166)(165,200,167)(211,238,218)(212,239,219)(213,240,220)(214,236,216)(215,237,217), (6,45)(7,41)(8,42)(9,43)(10,44)(11,230)(12,226)(13,227)(14,228)(15,229)(16,61)(17,62)(18,63)(19,64)(20,65)(21,220)(22,216)(23,217)(24,218)(25,219)(31,88)(32,89)(33,90)(34,86)(35,87)(36,223)(37,224)(38,225)(39,221)(40,222)(46,191)(47,192)(48,193)(49,194)(50,195)(51,77)(52,78)(53,79)(54,80)(55,76)(71,136)(72,137)(73,138)(74,139)(75,140)(81,118)(82,119)(83,120)(84,116)(85,117)(91,111)(92,112)(93,113)(94,114)(95,115)(96,148)(97,149)(98,150)(99,146)(100,147)(101,129)(102,130)(103,126)(104,127)(105,128)(106,121)(107,122)(108,123)(109,124)(110,125)(131,183)(132,184)(133,185)(134,181)(135,182)(151,158)(152,159)(153,160)(154,156)(155,157)(166,199)(167,200)(168,196)(169,197)(170,198)(171,210)(172,206)(173,207)(174,208)(175,209)(201,238)(202,239)(203,240)(204,236)(205,237)(211,231)(212,232)(213,233)(214,234)(215,235) );
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)], [(1,163,58,186),(2,164,59,187),(3,165,60,188),(4,161,56,189),(5,162,57,190),(6,133,90,170),(7,134,86,166),(8,135,87,167),(9,131,88,168),(10,132,89,169),(11,111,48,98),(12,112,49,99),(13,113,50,100),(14,114,46,96),(15,115,47,97),(16,152,53,206),(17,153,54,207),(18,154,55,208),(19,155,51,209),(20,151,52,210),(21,75,213,107),(22,71,214,108),(23,72,215,109),(24,73,211,110),(25,74,212,106),(26,141,66,178),(27,142,67,179),(28,143,68,180),(29,144,69,176),(30,145,70,177),(31,196,43,183),(32,197,44,184),(33,198,45,185),(34,199,41,181),(35,200,42,182),(36,81,223,118),(37,82,224,119),(38,83,225,120),(39,84,221,116),(40,85,222,117),(61,159,79,172),(62,160,80,173),(63,156,76,174),(64,157,77,175),(65,158,78,171),(91,193,150,230),(92,194,146,226),(93,195,147,227),(94,191,148,228),(95,192,149,229),(101,204,129,236),(102,205,130,237),(103,201,126,238),(104,202,127,239),(105,203,128,240),(121,219,139,232),(122,220,140,233),(123,216,136,234),(124,217,137,235),(125,218,138,231)], [(1,126,66,81),(2,127,67,82),(3,128,68,83),(4,129,69,84),(5,130,70,85),(6,110,52,91),(7,106,53,92),(8,107,54,93),(9,108,55,94),(10,109,51,95),(11,198,231,158),(12,199,232,159),(13,200,233,160),(14,196,234,156),(15,197,235,157),(16,146,86,74),(17,147,87,75),(18,148,88,71),(19,149,89,72),(20,150,90,73),(21,153,195,167),(22,154,191,168),(23,155,192,169),(24,151,193,170),(25,152,194,166),(26,118,58,103),(27,119,59,104),(28,120,60,105),(29,116,56,101),(30,117,57,102),(31,123,63,114),(32,124,64,115),(33,125,65,111),(34,121,61,112),(35,122,62,113),(36,163,201,178),(37,164,202,179),(38,165,203,180),(39,161,204,176),(40,162,205,177),(41,139,79,99),(42,140,80,100),(43,136,76,96),(44,137,77,97),(45,138,78,98),(46,183,216,174),(47,184,217,175),(48,185,218,171),(49,181,219,172),(50,182,220,173),(131,214,208,228),(132,215,209,229),(133,211,210,230),(134,212,206,226),(135,213,207,227),(141,223,186,238),(142,224,187,239),(143,225,188,240),(144,221,189,236),(145,222,190,237)], [(1,33,90),(2,34,86),(3,35,87),(4,31,88),(5,32,89),(6,58,45),(7,59,41),(8,60,42),(9,56,43),(10,57,44),(11,193,36),(12,194,37),(13,195,38),(14,191,39),(15,192,40),(16,67,61),(17,68,62),(18,69,63),(19,70,64),(20,66,65),(21,203,233),(22,204,234),(23,205,235),(24,201,231),(25,202,232),(26,78,52),(27,79,53),(28,80,54),(29,76,55),(30,77,51),(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),(81,111,150),(82,112,146),(83,113,147),(84,114,148),(85,115,149),(91,118,98),(92,119,99),(93,120,100),(94,116,96),(95,117,97),(101,136,108),(102,137,109),(103,138,110),(104,139,106),(105,140,107),(131,189,183),(132,190,184),(133,186,185),(134,187,181),(135,188,182),(141,171,210),(142,172,206),(143,173,207),(144,174,208),(145,175,209),(151,178,158),(152,179,159),(153,180,160),(154,176,156),(155,177,157),(161,196,168),(162,197,169),(163,198,170),(164,199,166),(165,200,167),(211,238,218),(212,239,219),(213,240,220),(214,236,216),(215,237,217)], [(6,45),(7,41),(8,42),(9,43),(10,44),(11,230),(12,226),(13,227),(14,228),(15,229),(16,61),(17,62),(18,63),(19,64),(20,65),(21,220),(22,216),(23,217),(24,218),(25,219),(31,88),(32,89),(33,90),(34,86),(35,87),(36,223),(37,224),(38,225),(39,221),(40,222),(46,191),(47,192),(48,193),(49,194),(50,195),(51,77),(52,78),(53,79),(54,80),(55,76),(71,136),(72,137),(73,138),(74,139),(75,140),(81,118),(82,119),(83,120),(84,116),(85,117),(91,111),(92,112),(93,113),(94,114),(95,115),(96,148),(97,149),(98,150),(99,146),(100,147),(101,129),(102,130),(103,126),(104,127),(105,128),(106,121),(107,122),(108,123),(109,124),(110,125),(131,183),(132,184),(133,185),(134,181),(135,182),(151,158),(152,159),(153,160),(154,156),(155,157),(166,199),(167,200),(168,196),(169,197),(170,198),(171,210),(172,206),(173,207),(174,208),(175,209),(201,238),(202,239),(203,240),(204,236),(205,237),(211,231),(212,232),(213,233),(214,234),(215,235)]])
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⋊C4⋊7S3 ►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⋊C4⋊7S3 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