metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (S3×C20)⋊7C4, (C4×Dic5)⋊9S3, (C4×S3)⋊3Dic5, C20.112(C4×S3), C60.154(C2×C4), C5⋊6(C42⋊2S3), (C12×Dic5)⋊8C2, (C2×C20).334D6, D6.5(C2×Dic5), C4.24(S3×Dic5), (C4×Dic15)⋊31C2, C6.26(C4○D20), C30.37(C4○D4), (C2×C12).338D10, D6⋊Dic5.15C2, (C2×C30).61C23, C6.Dic10⋊42C2, C12.29(C2×Dic5), C15⋊10(C42⋊C2), C10.30(C4○D12), C6.9(C22×Dic5), (C2×C60).236C22, C30.121(C22×C4), (C2×Dic5).162D6, Dic3.7(C2×Dic5), (C22×S3).68D10, (C2×Dic3).142D10, C2.2(D6.D10), C3⋊1(C23.21D10), (C6×Dic5).184C22, (C2×Dic15).190C22, (C10×Dic3).166C22, (S3×C2×C4).10D5, (S3×C2×C20).10C2, C10.117(S3×C2×C4), C2.11(C2×S3×Dic5), C22.34(C2×S3×D5), (C2×C4).239(S3×D5), (S3×C10).37(C2×C4), (S3×C2×C10).80C22, (C2×C6).73(C22×D5), (C2×C10).73(C22×S3), (C5×Dic3).45(C2×C4), SmallGroup(480,447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S3×C20)⋊7C4
G = < a,b,c,d | a20=b3=c2=d4=1, ab=ba, ac=ca, dad-1=a9, cbc=b-1, bd=db, dcd-1=a10c >
Subgroups: 556 in 152 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C30, C42⋊C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C4×Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C42⋊2S3, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C23.21D10, D6⋊Dic5, C6.Dic10, C12×Dic5, C4×Dic15, S3×C2×C20, (S3×C20)⋊7C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, Dic5, D10, C4×S3, C22×S3, C42⋊C2, C2×Dic5, C22×D5, S3×C2×C4, C4○D12, S3×D5, C4○D20, C22×Dic5, C42⋊2S3, S3×Dic5, C2×S3×D5, C23.21D10, D6.D10, C2×S3×Dic5, (S3×C20)⋊7C4
►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 47 85)(2 48 86)(3 49 87)(4 50 88)(5 51 89)(6 52 90)(7 53 91)(8 54 92)(9 55 93)(10 56 94)(11 57 95)(12 58 96)(13 59 97)(14 60 98)(15 41 99)(16 42 100)(17 43 81)(18 44 82)(19 45 83)(20 46 84)(21 66 140)(22 67 121)(23 68 122)(24 69 123)(25 70 124)(26 71 125)(27 72 126)(28 73 127)(29 74 128)(30 75 129)(31 76 130)(32 77 131)(33 78 132)(34 79 133)(35 80 134)(36 61 135)(37 62 136)(38 63 137)(39 64 138)(40 65 139)(101 195 204)(102 196 205)(103 197 206)(104 198 207)(105 199 208)(106 200 209)(107 181 210)(108 182 211)(109 183 212)(110 184 213)(111 185 214)(112 186 215)(113 187 216)(114 188 217)(115 189 218)(116 190 219)(117 191 220)(118 192 201)(119 193 202)(120 194 203)(141 166 227)(142 167 228)(143 168 229)(144 169 230)(145 170 231)(146 171 232)(147 172 233)(148 173 234)(149 174 235)(150 175 236)(151 176 237)(152 177 238)(153 178 239)(154 179 240)(155 180 221)(156 161 222)(157 162 223)(158 163 224)(159 164 225)(160 165 226)
(1 159)(2 160)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 239)(42 240)(43 221)(44 222)(45 223)(46 224)(47 225)(48 226)(49 227)(50 228)(51 229)(52 230)(53 231)(54 232)(55 233)(56 234)(57 235)(58 236)(59 237)(60 238)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 180)(82 161)(83 162)(84 163)(85 164)(86 165)(87 166)(88 167)(89 168)(90 169)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 178)(100 179)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 199)(133 200)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)
(1 132 159 189)(2 121 160 198)(3 130 141 187)(4 139 142 196)(5 128 143 185)(6 137 144 194)(7 126 145 183)(8 135 146 192)(9 124 147 181)(10 133 148 190)(11 122 149 199)(12 131 150 188)(13 140 151 197)(14 129 152 186)(15 138 153 195)(16 127 154 184)(17 136 155 193)(18 125 156 182)(19 134 157 191)(20 123 158 200)(21 176 206 59)(22 165 207 48)(23 174 208 57)(24 163 209 46)(25 172 210 55)(26 161 211 44)(27 170 212 53)(28 179 213 42)(29 168 214 51)(30 177 215 60)(31 166 216 49)(32 175 217 58)(33 164 218 47)(34 173 219 56)(35 162 220 45)(36 171 201 54)(37 180 202 43)(38 169 203 52)(39 178 204 41)(40 167 205 50)(61 232 118 92)(62 221 119 81)(63 230 120 90)(64 239 101 99)(65 228 102 88)(66 237 103 97)(67 226 104 86)(68 235 105 95)(69 224 106 84)(70 233 107 93)(71 222 108 82)(72 231 109 91)(73 240 110 100)(74 229 111 89)(75 238 112 98)(76 227 113 87)(77 236 114 96)(78 225 115 85)(79 234 116 94)(80 223 117 83)
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), (1,47,85)(2,48,86)(3,49,87)(4,50,88)(5,51,89)(6,52,90)(7,53,91)(8,54,92)(9,55,93)(10,56,94)(11,57,95)(12,58,96)(13,59,97)(14,60,98)(15,41,99)(16,42,100)(17,43,81)(18,44,82)(19,45,83)(20,46,84)(21,66,140)(22,67,121)(23,68,122)(24,69,123)(25,70,124)(26,71,125)(27,72,126)(28,73,127)(29,74,128)(30,75,129)(31,76,130)(32,77,131)(33,78,132)(34,79,133)(35,80,134)(36,61,135)(37,62,136)(38,63,137)(39,64,138)(40,65,139)(101,195,204)(102,196,205)(103,197,206)(104,198,207)(105,199,208)(106,200,209)(107,181,210)(108,182,211)(109,183,212)(110,184,213)(111,185,214)(112,186,215)(113,187,216)(114,188,217)(115,189,218)(116,190,219)(117,191,220)(118,192,201)(119,193,202)(120,194,203)(141,166,227)(142,167,228)(143,168,229)(144,169,230)(145,170,231)(146,171,232)(147,172,233)(148,173,234)(149,174,235)(150,175,236)(151,176,237)(152,177,238)(153,178,239)(154,179,240)(155,180,221)(156,161,222)(157,162,223)(158,163,224)(159,164,225)(160,165,226), (1,159)(2,160)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,239)(42,240)(43,221)(44,222)(45,223)(46,224)(47,225)(48,226)(49,227)(50,228)(51,229)(52,230)(53,231)(54,232)(55,233)(56,234)(57,235)(58,236)(59,237)(60,238)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,180)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,199)(133,200)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187), (1,132,159,189)(2,121,160,198)(3,130,141,187)(4,139,142,196)(5,128,143,185)(6,137,144,194)(7,126,145,183)(8,135,146,192)(9,124,147,181)(10,133,148,190)(11,122,149,199)(12,131,150,188)(13,140,151,197)(14,129,152,186)(15,138,153,195)(16,127,154,184)(17,136,155,193)(18,125,156,182)(19,134,157,191)(20,123,158,200)(21,176,206,59)(22,165,207,48)(23,174,208,57)(24,163,209,46)(25,172,210,55)(26,161,211,44)(27,170,212,53)(28,179,213,42)(29,168,214,51)(30,177,215,60)(31,166,216,49)(32,175,217,58)(33,164,218,47)(34,173,219,56)(35,162,220,45)(36,171,201,54)(37,180,202,43)(38,169,203,52)(39,178,204,41)(40,167,205,50)(61,232,118,92)(62,221,119,81)(63,230,120,90)(64,239,101,99)(65,228,102,88)(66,237,103,97)(67,226,104,86)(68,235,105,95)(69,224,106,84)(70,233,107,93)(71,222,108,82)(72,231,109,91)(73,240,110,100)(74,229,111,89)(75,238,112,98)(76,227,113,87)(77,236,114,96)(78,225,115,85)(79,234,116,94)(80,223,117,83)>;
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), (1,47,85)(2,48,86)(3,49,87)(4,50,88)(5,51,89)(6,52,90)(7,53,91)(8,54,92)(9,55,93)(10,56,94)(11,57,95)(12,58,96)(13,59,97)(14,60,98)(15,41,99)(16,42,100)(17,43,81)(18,44,82)(19,45,83)(20,46,84)(21,66,140)(22,67,121)(23,68,122)(24,69,123)(25,70,124)(26,71,125)(27,72,126)(28,73,127)(29,74,128)(30,75,129)(31,76,130)(32,77,131)(33,78,132)(34,79,133)(35,80,134)(36,61,135)(37,62,136)(38,63,137)(39,64,138)(40,65,139)(101,195,204)(102,196,205)(103,197,206)(104,198,207)(105,199,208)(106,200,209)(107,181,210)(108,182,211)(109,183,212)(110,184,213)(111,185,214)(112,186,215)(113,187,216)(114,188,217)(115,189,218)(116,190,219)(117,191,220)(118,192,201)(119,193,202)(120,194,203)(141,166,227)(142,167,228)(143,168,229)(144,169,230)(145,170,231)(146,171,232)(147,172,233)(148,173,234)(149,174,235)(150,175,236)(151,176,237)(152,177,238)(153,178,239)(154,179,240)(155,180,221)(156,161,222)(157,162,223)(158,163,224)(159,164,225)(160,165,226), (1,159)(2,160)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,239)(42,240)(43,221)(44,222)(45,223)(46,224)(47,225)(48,226)(49,227)(50,228)(51,229)(52,230)(53,231)(54,232)(55,233)(56,234)(57,235)(58,236)(59,237)(60,238)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,180)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,199)(133,200)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187), (1,132,159,189)(2,121,160,198)(3,130,141,187)(4,139,142,196)(5,128,143,185)(6,137,144,194)(7,126,145,183)(8,135,146,192)(9,124,147,181)(10,133,148,190)(11,122,149,199)(12,131,150,188)(13,140,151,197)(14,129,152,186)(15,138,153,195)(16,127,154,184)(17,136,155,193)(18,125,156,182)(19,134,157,191)(20,123,158,200)(21,176,206,59)(22,165,207,48)(23,174,208,57)(24,163,209,46)(25,172,210,55)(26,161,211,44)(27,170,212,53)(28,179,213,42)(29,168,214,51)(30,177,215,60)(31,166,216,49)(32,175,217,58)(33,164,218,47)(34,173,219,56)(35,162,220,45)(36,171,201,54)(37,180,202,43)(38,169,203,52)(39,178,204,41)(40,167,205,50)(61,232,118,92)(62,221,119,81)(63,230,120,90)(64,239,101,99)(65,228,102,88)(66,237,103,97)(67,226,104,86)(68,235,105,95)(69,224,106,84)(70,233,107,93)(71,222,108,82)(72,231,109,91)(73,240,110,100)(74,229,111,89)(75,238,112,98)(76,227,113,87)(77,236,114,96)(78,225,115,85)(79,234,116,94)(80,223,117,83) );
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)], [(1,47,85),(2,48,86),(3,49,87),(4,50,88),(5,51,89),(6,52,90),(7,53,91),(8,54,92),(9,55,93),(10,56,94),(11,57,95),(12,58,96),(13,59,97),(14,60,98),(15,41,99),(16,42,100),(17,43,81),(18,44,82),(19,45,83),(20,46,84),(21,66,140),(22,67,121),(23,68,122),(24,69,123),(25,70,124),(26,71,125),(27,72,126),(28,73,127),(29,74,128),(30,75,129),(31,76,130),(32,77,131),(33,78,132),(34,79,133),(35,80,134),(36,61,135),(37,62,136),(38,63,137),(39,64,138),(40,65,139),(101,195,204),(102,196,205),(103,197,206),(104,198,207),(105,199,208),(106,200,209),(107,181,210),(108,182,211),(109,183,212),(110,184,213),(111,185,214),(112,186,215),(113,187,216),(114,188,217),(115,189,218),(116,190,219),(117,191,220),(118,192,201),(119,193,202),(120,194,203),(141,166,227),(142,167,228),(143,168,229),(144,169,230),(145,170,231),(146,171,232),(147,172,233),(148,173,234),(149,174,235),(150,175,236),(151,176,237),(152,177,238),(153,178,239),(154,179,240),(155,180,221),(156,161,222),(157,162,223),(158,163,224),(159,164,225),(160,165,226)], [(1,159),(2,160),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,239),(42,240),(43,221),(44,222),(45,223),(46,224),(47,225),(48,226),(49,227),(50,228),(51,229),(52,230),(53,231),(54,232),(55,233),(56,234),(57,235),(58,236),(59,237),(60,238),(61,211),(62,212),(63,213),(64,214),(65,215),(66,216),(67,217),(68,218),(69,219),(70,220),(71,201),(72,202),(73,203),(74,204),(75,205),(76,206),(77,207),(78,208),(79,209),(80,210),(81,180),(82,161),(83,162),(84,163),(85,164),(86,165),(87,166),(88,167),(89,168),(90,169),(91,170),(92,171),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,178),(100,179),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(127,194),(128,195),(129,196),(130,197),(131,198),(132,199),(133,200),(134,181),(135,182),(136,183),(137,184),(138,185),(139,186),(140,187)], [(1,132,159,189),(2,121,160,198),(3,130,141,187),(4,139,142,196),(5,128,143,185),(6,137,144,194),(7,126,145,183),(8,135,146,192),(9,124,147,181),(10,133,148,190),(11,122,149,199),(12,131,150,188),(13,140,151,197),(14,129,152,186),(15,138,153,195),(16,127,154,184),(17,136,155,193),(18,125,156,182),(19,134,157,191),(20,123,158,200),(21,176,206,59),(22,165,207,48),(23,174,208,57),(24,163,209,46),(25,172,210,55),(26,161,211,44),(27,170,212,53),(28,179,213,42),(29,168,214,51),(30,177,215,60),(31,166,216,49),(32,175,217,58),(33,164,218,47),(34,173,219,56),(35,162,220,45),(36,171,201,54),(37,180,202,43),(38,169,203,52),(39,178,204,41),(40,167,205,50),(61,232,118,92),(62,221,119,81),(63,230,120,90),(64,239,101,99),(65,228,102,88),(66,237,103,97),(67,226,104,86),(68,235,105,95),(69,224,106,84),(70,233,107,93),(71,222,108,82),(72,231,109,91),(73,240,110,100),(74,229,111,89),(75,238,112,98),(76,227,113,87),(77,236,114,96),(78,225,115,85),(79,234,116,94),(80,223,117,83)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | D10 | C4×S3 | C4○D12 | C4○D20 | S3×D5 | S3×Dic5 | C2×S3×D5 | D6.D10 |
| kernel | (S3×C20)⋊7C4 | D6⋊Dic5 | C6.Dic10 | C12×Dic5 | C4×Dic15 | S3×C2×C20 | S3×C20 | C4×Dic5 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C10 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 8 | 2 | 2 | 2 | 4 | 8 | 16 | 2 | 4 | 2 | 8 |
Matrix representation of (S3×C20)⋊7C4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 53 | 0 |
| 0 | 0 | 0 | 19 | 23 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 47 | 0 | 0 |
| 0 | 22 | 59 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 22 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 46 | 60 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 53 | 3 |
| 0 | 0 | 0 | 40 | 8 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,53,19,0,0,0,0,23],[1,0,0,0,0,0,1,22,0,0,0,47,59,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,22,0,0,0,0,60,0,0,0,0,0,1,46,0,0,0,0,60],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,40,0,0,0,3,8] >;
(S3×C20)⋊7C4 in GAP, Magma, Sage, TeX
(S_3\times C_{20})\rtimes_7C_4 % in TeX
G:=Group("(S3xC20):7C4"); // GroupNames label
G:=SmallGroup(480,447);
// by ID
G=gap.SmallGroup(480,447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^3=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^10*c>;
// generators/relations