direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C22.D20, C4⋊Dic5⋊5C6, C10.5(C6×D4), C2.8(C6×D20), (C2×C6).26D20, C6.77(C2×D20), (C2×C30).81D4, D10⋊C4⋊6C6, C30.279(C2×D4), C22.4(C3×D20), C23.20(C6×D5), (C2×C12).231D10, (C22×Dic5)⋊5C6, (C22×C6).76D10, C30.231(C4○D4), (C2×C30).344C23, (C2×C60).266C22, C6.112(D4⋊2D5), C15⋊28(C22.D4), (C22×C30).102C22, (C6×Dic5).157C22, (C2×C4).7(C6×D5), (C5×C22⋊C4)⋊4C6, (C2×C20).3(C2×C6), C22⋊C4⋊6(C3×D5), (C2×C6×Dic5)⋊13C2, (C2×C10).4(C3×D4), (C2×C5⋊D4).5C6, C22.45(D5×C2×C6), (C3×C4⋊Dic5)⋊23C2, (C3×C22⋊C4)⋊14D5, C10.22(C3×C4○D4), (C6×C5⋊D4).12C2, (D5×C2×C6).79C22, (C15×C22⋊C4)⋊13C2, C5⋊2(C3×C22.D4), C2.10(C3×D4⋊2D5), (C2×Dic5).8(C2×C6), (C3×D10⋊C4)⋊18C2, (C22×D5).7(C2×C6), (C22×C10).21(C2×C6), (C2×C10).27(C22×C6), (C2×C6).340(C22×D5), SmallGroup(480,679)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.D20
G = < a,b,c,d,e | a3=b2=c2=d20=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 512 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C2×C30, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C3×C22.D4, C6×Dic5, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, C22×C30, C22.D20, C3×C4⋊Dic5, C3×D10⋊C4, C15×C22⋊C4, C2×C6×Dic5, C6×C5⋊D4, C3×C22.D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C22.D4, D20, C22×D5, C6×D4, C3×C4○D4, C6×D5, C2×D20, D4⋊2D5, C3×C22.D4, C3×D20, D5×C2×C6, C22.D20, C6×D20, C3×D4⋊2D5, C3×C22.D20
►On 240 points
Generators in S240
(1 51 21)(2 52 22)(3 53 23)(4 54 24)(5 55 25)(6 56 26)(7 57 27)(8 58 28)(9 59 29)(10 60 30)(11 41 31)(12 42 32)(13 43 33)(14 44 34)(15 45 35)(16 46 36)(17 47 37)(18 48 38)(19 49 39)(20 50 40)(61 234 110)(62 235 111)(63 236 112)(64 237 113)(65 238 114)(66 239 115)(67 240 116)(68 221 117)(69 222 118)(70 223 119)(71 224 120)(72 225 101)(73 226 102)(74 227 103)(75 228 104)(76 229 105)(77 230 106)(78 231 107)(79 232 108)(80 233 109)(81 191 122)(82 192 123)(83 193 124)(84 194 125)(85 195 126)(86 196 127)(87 197 128)(88 198 129)(89 199 130)(90 200 131)(91 181 132)(92 182 133)(93 183 134)(94 184 135)(95 185 136)(96 186 137)(97 187 138)(98 188 139)(99 189 140)(100 190 121)(141 203 161)(142 204 162)(143 205 163)(144 206 164)(145 207 165)(146 208 166)(147 209 167)(148 210 168)(149 211 169)(150 212 170)(151 213 171)(152 214 172)(153 215 173)(154 216 174)(155 217 175)(156 218 176)(157 219 177)(158 220 178)(159 201 179)(160 202 180)
(2 111)(4 113)(6 115)(8 117)(10 119)(12 101)(14 103)(16 105)(18 107)(20 109)(22 235)(24 237)(26 239)(28 221)(30 223)(32 225)(34 227)(36 229)(38 231)(40 233)(42 72)(44 74)(46 76)(48 78)(50 80)(52 62)(54 64)(56 66)(58 68)(60 70)(82 205)(84 207)(86 209)(88 211)(90 213)(92 215)(94 217)(96 219)(98 201)(100 203)(121 141)(123 143)(125 145)(127 147)(129 149)(131 151)(133 153)(135 155)(137 157)(139 159)(161 190)(163 192)(165 194)(167 196)(169 198)(171 200)(173 182)(175 184)(177 186)(179 188)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 234)(22 235)(23 236)(24 237)(25 238)(26 239)(27 240)(28 221)(29 222)(30 223)(31 224)(32 225)(33 226)(34 227)(35 228)(36 229)(37 230)(38 231)(39 232)(40 233)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 204)(82 205)(83 206)(84 207)(85 208)(86 209)(87 210)(88 211)(89 212)(90 213)(91 214)(92 215)(93 216)(94 217)(95 218)(96 219)(97 220)(98 201)(99 202)(100 203)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(161 190)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 197)(169 198)(170 199)(171 200)(172 181)(173 182)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)
(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 99 110 202)(2 201 111 98)(3 97 112 220)(4 219 113 96)(5 95 114 218)(6 217 115 94)(7 93 116 216)(8 215 117 92)(9 91 118 214)(10 213 119 90)(11 89 120 212)(12 211 101 88)(13 87 102 210)(14 209 103 86)(15 85 104 208)(16 207 105 84)(17 83 106 206)(18 205 107 82)(19 81 108 204)(20 203 109 100)(21 140 234 160)(22 159 235 139)(23 138 236 158)(24 157 237 137)(25 136 238 156)(26 155 239 135)(27 134 240 154)(28 153 221 133)(29 132 222 152)(30 151 223 131)(31 130 224 150)(32 149 225 129)(33 128 226 148)(34 147 227 127)(35 126 228 146)(36 145 229 125)(37 124 230 144)(38 143 231 123)(39 122 232 142)(40 141 233 121)(41 199 71 170)(42 169 72 198)(43 197 73 168)(44 167 74 196)(45 195 75 166)(46 165 76 194)(47 193 77 164)(48 163 78 192)(49 191 79 162)(50 161 80 190)(51 189 61 180)(52 179 62 188)(53 187 63 178)(54 177 64 186)(55 185 65 176)(56 175 66 184)(57 183 67 174)(58 173 68 182)(59 181 69 172)(60 171 70 200)
G:=sub<Sym(240)| (1,51,21)(2,52,22)(3,53,23)(4,54,24)(5,55,25)(6,56,26)(7,57,27)(8,58,28)(9,59,29)(10,60,30)(11,41,31)(12,42,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(61,234,110)(62,235,111)(63,236,112)(64,237,113)(65,238,114)(66,239,115)(67,240,116)(68,221,117)(69,222,118)(70,223,119)(71,224,120)(72,225,101)(73,226,102)(74,227,103)(75,228,104)(76,229,105)(77,230,106)(78,231,107)(79,232,108)(80,233,109)(81,191,122)(82,192,123)(83,193,124)(84,194,125)(85,195,126)(86,196,127)(87,197,128)(88,198,129)(89,199,130)(90,200,131)(91,181,132)(92,182,133)(93,183,134)(94,184,135)(95,185,136)(96,186,137)(97,187,138)(98,188,139)(99,189,140)(100,190,121)(141,203,161)(142,204,162)(143,205,163)(144,206,164)(145,207,165)(146,208,166)(147,209,167)(148,210,168)(149,211,169)(150,212,170)(151,213,171)(152,214,172)(153,215,173)(154,216,174)(155,217,175)(156,218,176)(157,219,177)(158,220,178)(159,201,179)(160,202,180), (2,111)(4,113)(6,115)(8,117)(10,119)(12,101)(14,103)(16,105)(18,107)(20,109)(22,235)(24,237)(26,239)(28,221)(30,223)(32,225)(34,227)(36,229)(38,231)(40,233)(42,72)(44,74)(46,76)(48,78)(50,80)(52,62)(54,64)(56,66)(58,68)(60,70)(82,205)(84,207)(86,209)(88,211)(90,213)(92,215)(94,217)(96,219)(98,201)(100,203)(121,141)(123,143)(125,145)(127,147)(129,149)(131,151)(133,153)(135,155)(137,157)(139,159)(161,190)(163,192)(165,194)(167,196)(169,198)(171,200)(173,182)(175,184)(177,186)(179,188), (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,234)(22,235)(23,236)(24,237)(25,238)(26,239)(27,240)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,201)(99,202)(100,203)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,199)(171,200)(172,181)(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189), (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), (1,99,110,202)(2,201,111,98)(3,97,112,220)(4,219,113,96)(5,95,114,218)(6,217,115,94)(7,93,116,216)(8,215,117,92)(9,91,118,214)(10,213,119,90)(11,89,120,212)(12,211,101,88)(13,87,102,210)(14,209,103,86)(15,85,104,208)(16,207,105,84)(17,83,106,206)(18,205,107,82)(19,81,108,204)(20,203,109,100)(21,140,234,160)(22,159,235,139)(23,138,236,158)(24,157,237,137)(25,136,238,156)(26,155,239,135)(27,134,240,154)(28,153,221,133)(29,132,222,152)(30,151,223,131)(31,130,224,150)(32,149,225,129)(33,128,226,148)(34,147,227,127)(35,126,228,146)(36,145,229,125)(37,124,230,144)(38,143,231,123)(39,122,232,142)(40,141,233,121)(41,199,71,170)(42,169,72,198)(43,197,73,168)(44,167,74,196)(45,195,75,166)(46,165,76,194)(47,193,77,164)(48,163,78,192)(49,191,79,162)(50,161,80,190)(51,189,61,180)(52,179,62,188)(53,187,63,178)(54,177,64,186)(55,185,65,176)(56,175,66,184)(57,183,67,174)(58,173,68,182)(59,181,69,172)(60,171,70,200)>;
G:=Group( (1,51,21)(2,52,22)(3,53,23)(4,54,24)(5,55,25)(6,56,26)(7,57,27)(8,58,28)(9,59,29)(10,60,30)(11,41,31)(12,42,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(61,234,110)(62,235,111)(63,236,112)(64,237,113)(65,238,114)(66,239,115)(67,240,116)(68,221,117)(69,222,118)(70,223,119)(71,224,120)(72,225,101)(73,226,102)(74,227,103)(75,228,104)(76,229,105)(77,230,106)(78,231,107)(79,232,108)(80,233,109)(81,191,122)(82,192,123)(83,193,124)(84,194,125)(85,195,126)(86,196,127)(87,197,128)(88,198,129)(89,199,130)(90,200,131)(91,181,132)(92,182,133)(93,183,134)(94,184,135)(95,185,136)(96,186,137)(97,187,138)(98,188,139)(99,189,140)(100,190,121)(141,203,161)(142,204,162)(143,205,163)(144,206,164)(145,207,165)(146,208,166)(147,209,167)(148,210,168)(149,211,169)(150,212,170)(151,213,171)(152,214,172)(153,215,173)(154,216,174)(155,217,175)(156,218,176)(157,219,177)(158,220,178)(159,201,179)(160,202,180), (2,111)(4,113)(6,115)(8,117)(10,119)(12,101)(14,103)(16,105)(18,107)(20,109)(22,235)(24,237)(26,239)(28,221)(30,223)(32,225)(34,227)(36,229)(38,231)(40,233)(42,72)(44,74)(46,76)(48,78)(50,80)(52,62)(54,64)(56,66)(58,68)(60,70)(82,205)(84,207)(86,209)(88,211)(90,213)(92,215)(94,217)(96,219)(98,201)(100,203)(121,141)(123,143)(125,145)(127,147)(129,149)(131,151)(133,153)(135,155)(137,157)(139,159)(161,190)(163,192)(165,194)(167,196)(169,198)(171,200)(173,182)(175,184)(177,186)(179,188), (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,234)(22,235)(23,236)(24,237)(25,238)(26,239)(27,240)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,201)(99,202)(100,203)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,199)(171,200)(172,181)(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189), (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), (1,99,110,202)(2,201,111,98)(3,97,112,220)(4,219,113,96)(5,95,114,218)(6,217,115,94)(7,93,116,216)(8,215,117,92)(9,91,118,214)(10,213,119,90)(11,89,120,212)(12,211,101,88)(13,87,102,210)(14,209,103,86)(15,85,104,208)(16,207,105,84)(17,83,106,206)(18,205,107,82)(19,81,108,204)(20,203,109,100)(21,140,234,160)(22,159,235,139)(23,138,236,158)(24,157,237,137)(25,136,238,156)(26,155,239,135)(27,134,240,154)(28,153,221,133)(29,132,222,152)(30,151,223,131)(31,130,224,150)(32,149,225,129)(33,128,226,148)(34,147,227,127)(35,126,228,146)(36,145,229,125)(37,124,230,144)(38,143,231,123)(39,122,232,142)(40,141,233,121)(41,199,71,170)(42,169,72,198)(43,197,73,168)(44,167,74,196)(45,195,75,166)(46,165,76,194)(47,193,77,164)(48,163,78,192)(49,191,79,162)(50,161,80,190)(51,189,61,180)(52,179,62,188)(53,187,63,178)(54,177,64,186)(55,185,65,176)(56,175,66,184)(57,183,67,174)(58,173,68,182)(59,181,69,172)(60,171,70,200) );
G=PermutationGroup([[(1,51,21),(2,52,22),(3,53,23),(4,54,24),(5,55,25),(6,56,26),(7,57,27),(8,58,28),(9,59,29),(10,60,30),(11,41,31),(12,42,32),(13,43,33),(14,44,34),(15,45,35),(16,46,36),(17,47,37),(18,48,38),(19,49,39),(20,50,40),(61,234,110),(62,235,111),(63,236,112),(64,237,113),(65,238,114),(66,239,115),(67,240,116),(68,221,117),(69,222,118),(70,223,119),(71,224,120),(72,225,101),(73,226,102),(74,227,103),(75,228,104),(76,229,105),(77,230,106),(78,231,107),(79,232,108),(80,233,109),(81,191,122),(82,192,123),(83,193,124),(84,194,125),(85,195,126),(86,196,127),(87,197,128),(88,198,129),(89,199,130),(90,200,131),(91,181,132),(92,182,133),(93,183,134),(94,184,135),(95,185,136),(96,186,137),(97,187,138),(98,188,139),(99,189,140),(100,190,121),(141,203,161),(142,204,162),(143,205,163),(144,206,164),(145,207,165),(146,208,166),(147,209,167),(148,210,168),(149,211,169),(150,212,170),(151,213,171),(152,214,172),(153,215,173),(154,216,174),(155,217,175),(156,218,176),(157,219,177),(158,220,178),(159,201,179),(160,202,180)], [(2,111),(4,113),(6,115),(8,117),(10,119),(12,101),(14,103),(16,105),(18,107),(20,109),(22,235),(24,237),(26,239),(28,221),(30,223),(32,225),(34,227),(36,229),(38,231),(40,233),(42,72),(44,74),(46,76),(48,78),(50,80),(52,62),(54,64),(56,66),(58,68),(60,70),(82,205),(84,207),(86,209),(88,211),(90,213),(92,215),(94,217),(96,219),(98,201),(100,203),(121,141),(123,143),(125,145),(127,147),(129,149),(131,151),(133,153),(135,155),(137,157),(139,159),(161,190),(163,192),(165,194),(167,196),(169,198),(171,200),(173,182),(175,184),(177,186),(179,188)], [(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,234),(22,235),(23,236),(24,237),(25,238),(26,239),(27,240),(28,221),(29,222),(30,223),(31,224),(32,225),(33,226),(34,227),(35,228),(36,229),(37,230),(38,231),(39,232),(40,233),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(81,204),(82,205),(83,206),(84,207),(85,208),(86,209),(87,210),(88,211),(89,212),(90,213),(91,214),(92,215),(93,216),(94,217),(95,218),(96,219),(97,220),(98,201),(99,202),(100,203),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160),(161,190),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,197),(169,198),(170,199),(171,200),(172,181),(173,182),(174,183),(175,184),(176,185),(177,186),(178,187),(179,188),(180,189)], [(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)], [(1,99,110,202),(2,201,111,98),(3,97,112,220),(4,219,113,96),(5,95,114,218),(6,217,115,94),(7,93,116,216),(8,215,117,92),(9,91,118,214),(10,213,119,90),(11,89,120,212),(12,211,101,88),(13,87,102,210),(14,209,103,86),(15,85,104,208),(16,207,105,84),(17,83,106,206),(18,205,107,82),(19,81,108,204),(20,203,109,100),(21,140,234,160),(22,159,235,139),(23,138,236,158),(24,157,237,137),(25,136,238,156),(26,155,239,135),(27,134,240,154),(28,153,221,133),(29,132,222,152),(30,151,223,131),(31,130,224,150),(32,149,225,129),(33,128,226,148),(34,147,227,127),(35,126,228,146),(36,145,229,125),(37,124,230,144),(38,143,231,123),(39,122,232,142),(40,141,233,121),(41,199,71,170),(42,169,72,198),(43,197,73,168),(44,167,74,196),(45,195,75,166),(46,165,76,194),(47,193,77,164),(48,163,78,192),(49,191,79,162),(50,161,80,190),(51,189,61,180),(52,179,62,188),(53,187,63,178),(54,177,64,186),(55,185,65,176),(56,175,66,184),(57,183,67,174),(58,173,68,182),(59,181,69,172),(60,171,70,200)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | D20 | C3×C4○D4 | C6×D5 | C6×D5 | C3×D20 | D4⋊2D5 | C3×D4⋊2D5 |
| kernel | C3×C22.D20 | C3×C4⋊Dic5 | C3×D10⋊C4 | C15×C22⋊C4 | C2×C6×Dic5 | C6×C5⋊D4 | C22.D20 | C4⋊Dic5 | D10⋊C4 | C5×C22⋊C4 | C22×Dic5 | C2×C5⋊D4 | C2×C30 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 4 | 8 |
Matrix representation of C3×C22.D20 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 20 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 2 | 29 | 0 | 0 |
| 32 | 7 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 60 |
| 33 | 25 | 0 | 0 |
| 37 | 28 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 24 | 11 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[2,32,0,0,29,7,0,0,0,0,1,0,0,0,6,60],[33,37,0,0,25,28,0,0,0,0,50,24,0,0,0,11] >;
C3×C22.D20 in GAP, Magma, Sage, TeX
C_3\times C_2^2.D_{20} % in TeX
G:=Group("C3xC2^2.D20"); // GroupNames label
G:=SmallGroup(480,679);
// by ID
G=gap.SmallGroup(480,679);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,555,394,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^20=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations