direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20×C3⋊D4, C60⋊34D4, C3⋊4(D4×C20), C12⋊8(C5×D4), C15⋊42(C4×D4), D6⋊4(C2×C20), D6⋊C4⋊18C10, (C22×C20)⋊6S3, C22⋊3(S3×C20), C6.41(D4×C10), (C22×C12)⋊9C10, (C22×C60)⋊23C2, Dic3⋊2(C2×C20), C30.424(C2×D4), (C2×C20).376D6, Dic3⋊C4⋊18C10, (C4×Dic3)⋊16C10, (Dic3×C20)⋊34C2, C6.19(C22×C20), C23.27(S3×C10), C30.210(C4○D4), C6.D4⋊14C10, (C2×C30).425C23, (C2×C60).456C22, C30.210(C22×C4), (C22×C10).124D6, C10.124(C4○D12), (C22×C30).176C22, (C10×Dic3).226C22, (S3×C2×C4)⋊14C10, (S3×C2×C20)⋊30C2, (C2×C6)⋊5(C2×C20), C2.20(S3×C2×C20), (C2×C10)⋊19(C4×S3), (C2×C30)⋊37(C2×C4), (C5×D6⋊C4)⋊40C2, C10.147(S3×C2×C4), (C22×C4)⋊4(C5×S3), C2.5(C5×C4○D12), C6.15(C5×C4○D4), C2.3(C10×C3⋊D4), (S3×C10)⋊26(C2×C4), (C2×C3⋊D4).7C10, C22.24(S3×C2×C10), (C2×C4).103(S3×C10), (C2×C12).75(C2×C10), (C5×Dic3⋊C4)⋊40C2, (C5×Dic3)⋊18(C2×C4), (C10×C3⋊D4).14C2, C10.126(C2×C3⋊D4), (C5×C6.D4)⋊30C2, (S3×C2×C10).116C22, (C2×C6).46(C22×C10), (C22×C6).38(C2×C10), (C22×S3).25(C2×C10), (C2×C10).359(C22×S3), (C2×Dic3).34(C2×C10), SmallGroup(480,807)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20×C3⋊D4
G = < a,b,c,d | a20=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 388 in 188 conjugacy classes, 90 normal (58 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C4×C3⋊D4, S3×C20, C10×Dic3, C5×C3⋊D4, C2×C60, C2×C60, S3×C2×C10, C22×C30, D4×C20, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C5×C6.D4, S3×C2×C20, C10×C3⋊D4, C22×C60, C20×C3⋊D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×S3, C3⋊D4, C22×S3, C5×S3, C4×D4, C2×C20, C5×D4, C22×C10, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×C10, C22×C20, D4×C10, C5×C4○D4, C4×C3⋊D4, S3×C20, C5×C3⋊D4, S3×C2×C10, D4×C20, S3×C2×C20, C5×C4○D12, C10×C3⋊D4, C20×C3⋊D4
►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)
(1 192 233)(2 193 234)(3 194 235)(4 195 236)(5 196 237)(6 197 238)(7 198 239)(8 199 240)(9 200 221)(10 181 222)(11 182 223)(12 183 224)(13 184 225)(14 185 226)(15 186 227)(16 187 228)(17 188 229)(18 189 230)(19 190 231)(20 191 232)(21 92 135)(22 93 136)(23 94 137)(24 95 138)(25 96 139)(26 97 140)(27 98 121)(28 99 122)(29 100 123)(30 81 124)(31 82 125)(32 83 126)(33 84 127)(34 85 128)(35 86 129)(36 87 130)(37 88 131)(38 89 132)(39 90 133)(40 91 134)(41 118 220)(42 119 201)(43 120 202)(44 101 203)(45 102 204)(46 103 205)(47 104 206)(48 105 207)(49 106 208)(50 107 209)(51 108 210)(52 109 211)(53 110 212)(54 111 213)(55 112 214)(56 113 215)(57 114 216)(58 115 217)(59 116 218)(60 117 219)(61 144 173)(62 145 174)(63 146 175)(64 147 176)(65 148 177)(66 149 178)(67 150 179)(68 151 180)(69 152 161)(70 153 162)(71 154 163)(72 155 164)(73 156 165)(74 157 166)(75 158 167)(76 159 168)(77 160 169)(78 141 170)(79 142 171)(80 143 172)
(1 52 135 70)(2 53 136 71)(3 54 137 72)(4 55 138 73)(5 56 139 74)(6 57 140 75)(7 58 121 76)(8 59 122 77)(9 60 123 78)(10 41 124 79)(11 42 125 80)(12 43 126 61)(13 44 127 62)(14 45 128 63)(15 46 129 64)(16 47 130 65)(17 48 131 66)(18 49 132 67)(19 50 133 68)(20 51 134 69)(21 162 192 211)(22 163 193 212)(23 164 194 213)(24 165 195 214)(25 166 196 215)(26 167 197 216)(27 168 198 217)(28 169 199 218)(29 170 200 219)(30 171 181 220)(31 172 182 201)(32 173 183 202)(33 174 184 203)(34 175 185 204)(35 176 186 205)(36 177 187 206)(37 178 188 207)(38 179 189 208)(39 180 190 209)(40 161 191 210)(81 142 222 118)(82 143 223 119)(83 144 224 120)(84 145 225 101)(85 146 226 102)(86 147 227 103)(87 148 228 104)(88 149 229 105)(89 150 230 106)(90 151 231 107)(91 152 232 108)(92 153 233 109)(93 154 234 110)(94 155 235 111)(95 156 236 112)(96 157 237 113)(97 158 238 114)(98 159 239 115)(99 160 240 116)(100 141 221 117)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 81)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(101 164)(102 165)(103 166)(104 167)(105 168)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 176)(114 177)(115 178)(116 179)(117 180)(118 161)(119 162)(120 163)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 209)(142 210)(143 211)(144 212)(145 213)(146 214)(147 215)(148 216)(149 217)(150 218)(151 219)(152 220)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(181 232)(182 233)(183 234)(184 235)(185 236)(186 237)(187 238)(188 239)(189 240)(190 221)(191 222)(192 223)(193 224)(194 225)(195 226)(196 227)(197 228)(198 229)(199 230)(200 231)
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), (1,192,233)(2,193,234)(3,194,235)(4,195,236)(5,196,237)(6,197,238)(7,198,239)(8,199,240)(9,200,221)(10,181,222)(11,182,223)(12,183,224)(13,184,225)(14,185,226)(15,186,227)(16,187,228)(17,188,229)(18,189,230)(19,190,231)(20,191,232)(21,92,135)(22,93,136)(23,94,137)(24,95,138)(25,96,139)(26,97,140)(27,98,121)(28,99,122)(29,100,123)(30,81,124)(31,82,125)(32,83,126)(33,84,127)(34,85,128)(35,86,129)(36,87,130)(37,88,131)(38,89,132)(39,90,133)(40,91,134)(41,118,220)(42,119,201)(43,120,202)(44,101,203)(45,102,204)(46,103,205)(47,104,206)(48,105,207)(49,106,208)(50,107,209)(51,108,210)(52,109,211)(53,110,212)(54,111,213)(55,112,214)(56,113,215)(57,114,216)(58,115,217)(59,116,218)(60,117,219)(61,144,173)(62,145,174)(63,146,175)(64,147,176)(65,148,177)(66,149,178)(67,150,179)(68,151,180)(69,152,161)(70,153,162)(71,154,163)(72,155,164)(73,156,165)(74,157,166)(75,158,167)(76,159,168)(77,160,169)(78,141,170)(79,142,171)(80,143,172), (1,52,135,70)(2,53,136,71)(3,54,137,72)(4,55,138,73)(5,56,139,74)(6,57,140,75)(7,58,121,76)(8,59,122,77)(9,60,123,78)(10,41,124,79)(11,42,125,80)(12,43,126,61)(13,44,127,62)(14,45,128,63)(15,46,129,64)(16,47,130,65)(17,48,131,66)(18,49,132,67)(19,50,133,68)(20,51,134,69)(21,162,192,211)(22,163,193,212)(23,164,194,213)(24,165,195,214)(25,166,196,215)(26,167,197,216)(27,168,198,217)(28,169,199,218)(29,170,200,219)(30,171,181,220)(31,172,182,201)(32,173,183,202)(33,174,184,203)(34,175,185,204)(35,176,186,205)(36,177,187,206)(37,178,188,207)(38,179,189,208)(39,180,190,209)(40,161,191,210)(81,142,222,118)(82,143,223,119)(83,144,224,120)(84,145,225,101)(85,146,226,102)(86,147,227,103)(87,148,228,104)(88,149,229,105)(89,150,230,106)(90,151,231,107)(91,152,232,108)(92,153,233,109)(93,154,234,110)(94,155,235,111)(95,156,236,112)(96,157,237,113)(97,158,238,114)(98,159,239,115)(99,160,240,116)(100,141,221,117), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,161)(119,162)(120,163)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,209)(142,210)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,221)(191,222)(192,223)(193,224)(194,225)(195,226)(196,227)(197,228)(198,229)(199,230)(200,231)>;
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), (1,192,233)(2,193,234)(3,194,235)(4,195,236)(5,196,237)(6,197,238)(7,198,239)(8,199,240)(9,200,221)(10,181,222)(11,182,223)(12,183,224)(13,184,225)(14,185,226)(15,186,227)(16,187,228)(17,188,229)(18,189,230)(19,190,231)(20,191,232)(21,92,135)(22,93,136)(23,94,137)(24,95,138)(25,96,139)(26,97,140)(27,98,121)(28,99,122)(29,100,123)(30,81,124)(31,82,125)(32,83,126)(33,84,127)(34,85,128)(35,86,129)(36,87,130)(37,88,131)(38,89,132)(39,90,133)(40,91,134)(41,118,220)(42,119,201)(43,120,202)(44,101,203)(45,102,204)(46,103,205)(47,104,206)(48,105,207)(49,106,208)(50,107,209)(51,108,210)(52,109,211)(53,110,212)(54,111,213)(55,112,214)(56,113,215)(57,114,216)(58,115,217)(59,116,218)(60,117,219)(61,144,173)(62,145,174)(63,146,175)(64,147,176)(65,148,177)(66,149,178)(67,150,179)(68,151,180)(69,152,161)(70,153,162)(71,154,163)(72,155,164)(73,156,165)(74,157,166)(75,158,167)(76,159,168)(77,160,169)(78,141,170)(79,142,171)(80,143,172), (1,52,135,70)(2,53,136,71)(3,54,137,72)(4,55,138,73)(5,56,139,74)(6,57,140,75)(7,58,121,76)(8,59,122,77)(9,60,123,78)(10,41,124,79)(11,42,125,80)(12,43,126,61)(13,44,127,62)(14,45,128,63)(15,46,129,64)(16,47,130,65)(17,48,131,66)(18,49,132,67)(19,50,133,68)(20,51,134,69)(21,162,192,211)(22,163,193,212)(23,164,194,213)(24,165,195,214)(25,166,196,215)(26,167,197,216)(27,168,198,217)(28,169,199,218)(29,170,200,219)(30,171,181,220)(31,172,182,201)(32,173,183,202)(33,174,184,203)(34,175,185,204)(35,176,186,205)(36,177,187,206)(37,178,188,207)(38,179,189,208)(39,180,190,209)(40,161,191,210)(81,142,222,118)(82,143,223,119)(83,144,224,120)(84,145,225,101)(85,146,226,102)(86,147,227,103)(87,148,228,104)(88,149,229,105)(89,150,230,106)(90,151,231,107)(91,152,232,108)(92,153,233,109)(93,154,234,110)(94,155,235,111)(95,156,236,112)(96,157,237,113)(97,158,238,114)(98,159,239,115)(99,160,240,116)(100,141,221,117), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,81)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,161)(119,162)(120,163)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,209)(142,210)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,221)(191,222)(192,223)(193,224)(194,225)(195,226)(196,227)(197,228)(198,229)(199,230)(200,231) );
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)], [(1,192,233),(2,193,234),(3,194,235),(4,195,236),(5,196,237),(6,197,238),(7,198,239),(8,199,240),(9,200,221),(10,181,222),(11,182,223),(12,183,224),(13,184,225),(14,185,226),(15,186,227),(16,187,228),(17,188,229),(18,189,230),(19,190,231),(20,191,232),(21,92,135),(22,93,136),(23,94,137),(24,95,138),(25,96,139),(26,97,140),(27,98,121),(28,99,122),(29,100,123),(30,81,124),(31,82,125),(32,83,126),(33,84,127),(34,85,128),(35,86,129),(36,87,130),(37,88,131),(38,89,132),(39,90,133),(40,91,134),(41,118,220),(42,119,201),(43,120,202),(44,101,203),(45,102,204),(46,103,205),(47,104,206),(48,105,207),(49,106,208),(50,107,209),(51,108,210),(52,109,211),(53,110,212),(54,111,213),(55,112,214),(56,113,215),(57,114,216),(58,115,217),(59,116,218),(60,117,219),(61,144,173),(62,145,174),(63,146,175),(64,147,176),(65,148,177),(66,149,178),(67,150,179),(68,151,180),(69,152,161),(70,153,162),(71,154,163),(72,155,164),(73,156,165),(74,157,166),(75,158,167),(76,159,168),(77,160,169),(78,141,170),(79,142,171),(80,143,172)], [(1,52,135,70),(2,53,136,71),(3,54,137,72),(4,55,138,73),(5,56,139,74),(6,57,140,75),(7,58,121,76),(8,59,122,77),(9,60,123,78),(10,41,124,79),(11,42,125,80),(12,43,126,61),(13,44,127,62),(14,45,128,63),(15,46,129,64),(16,47,130,65),(17,48,131,66),(18,49,132,67),(19,50,133,68),(20,51,134,69),(21,162,192,211),(22,163,193,212),(23,164,194,213),(24,165,195,214),(25,166,196,215),(26,167,197,216),(27,168,198,217),(28,169,199,218),(29,170,200,219),(30,171,181,220),(31,172,182,201),(32,173,183,202),(33,174,184,203),(34,175,185,204),(35,176,186,205),(36,177,187,206),(37,178,188,207),(38,179,189,208),(39,180,190,209),(40,161,191,210),(81,142,222,118),(82,143,223,119),(83,144,224,120),(84,145,225,101),(85,146,226,102),(86,147,227,103),(87,148,228,104),(88,149,229,105),(89,150,230,106),(90,151,231,107),(91,152,232,108),(92,153,233,109),(93,154,234,110),(94,155,235,111),(95,156,236,112),(96,157,237,113),(97,158,238,114),(98,159,239,115),(99,160,240,116),(100,141,221,117)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,81),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(101,164),(102,165),(103,166),(104,167),(105,168),(106,169),(107,170),(108,171),(109,172),(110,173),(111,174),(112,175),(113,176),(114,177),(115,178),(116,179),(117,180),(118,161),(119,162),(120,163),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,209),(142,210),(143,211),(144,212),(145,213),(146,214),(147,215),(148,216),(149,217),(150,218),(151,219),(152,220),(153,201),(154,202),(155,203),(156,204),(157,205),(158,206),(159,207),(160,208),(181,232),(182,233),(183,234),(184,235),(185,236),(186,237),(187,238),(188,239),(189,240),(190,221),(191,222),(192,223),(193,224),(194,225),(195,226),(196,227),(197,228),(198,229),(199,230),(200,231)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AV | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 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 | 2 | 2 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C5×S3 | C5×D4 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | S3×C20 | C5×C4○D12 |
| kernel | C20×C3⋊D4 | Dic3×C20 | C5×Dic3⋊C4 | C5×D6⋊C4 | C5×C6.D4 | S3×C2×C20 | C10×C3⋊D4 | C22×C60 | C5×C3⋊D4 | C4×C3⋊D4 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C6.D4 | S3×C2×C4 | C2×C3⋊D4 | C22×C12 | C3⋊D4 | C22×C20 | C60 | C2×C20 | C22×C10 | C30 | C20 | C2×C10 | C22×C4 | C12 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 32 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C20×C3⋊D4 ►in GL3(𝔽61) generated by
| 11 | 0 | 0 |
| 0 | 41 | 0 |
| 0 | 0 | 41 |
| 1 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 60 |
| 60 | 0 | 0 |
| 0 | 18 | 52 |
| 0 | 9 | 43 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(61))| [11,0,0,0,41,0,0,0,41],[1,0,0,0,0,1,0,60,60],[60,0,0,0,18,9,0,52,43],[1,0,0,0,0,1,0,1,0] >;
C20×C3⋊D4 in GAP, Magma, Sage, TeX
C_{20}\times C_3\rtimes D_4 % in TeX
G:=Group("C20xC3:D4"); // GroupNames label
G:=SmallGroup(480,807);
// by ID
G=gap.SmallGroup(480,807);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations