direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C3⋊D20, D30⋊9C23, C30.47C24, C6⋊3(C2×D20), C30⋊5(C2×D4), (C2×C6)⋊9D20, (C2×C30)⋊14D4, C15⋊6(C22×D4), C3⋊3(C22×D20), (C6×D5)⋊7C23, (C23×D5)⋊7S3, (C22×D5)⋊15D6, D10⋊7(C22×S3), (C23×D15)⋊8C2, C6.47(C23×D5), C23.71(S3×D5), (C2×Dic3)⋊24D10, C10.47(S3×C23), (C5×Dic3)⋊8C23, (C22×Dic3)⋊9D5, Dic3⋊5(C22×D5), (C2×C30).250C23, (C22×C6).102D10, (C22×C10).119D6, (C10×Dic3)⋊31C22, (C22×D15)⋊20C22, (C22×C30).88C22, C10⋊1(C2×C3⋊D4), (D5×C22×C6)⋊4C2, C5⋊1(C22×C3⋊D4), (D5×C2×C6)⋊18C22, C2.47(C22×S3×D5), (Dic3×C2×C10)⋊11C2, (C2×C10)⋊13(C3⋊D4), C22.110(C2×S3×D5), (C2×C6).256(C22×D5), (C2×C10).254(C22×S3), SmallGroup(480,1119)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2684 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×6], Dic3 [×4], D6 [×16], C2×C6 [×7], C2×C6 [×16], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], D10 [×4], D10 [×28], C2×C10 [×7], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×10], C22×C6, C22×C6 [×10], C3×D5 [×4], D15 [×4], C30, C30 [×6], C22×D4, D20 [×16], C2×C20 [×6], C22×D5 [×6], C22×D5 [×14], C22×C10, C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C5×Dic3 [×4], C6×D5 [×4], C6×D5 [×12], D30 [×4], D30 [×12], C2×C30 [×7], C2×D20 [×12], C22×C20, C23×D5, C23×D5, C22×C3⋊D4, C3⋊D20 [×16], C10×Dic3 [×6], D5×C2×C6 [×6], D5×C2×C6 [×4], C22×D15 [×6], C22×D15 [×4], C22×C30, C22×D20, C2×C3⋊D20 [×12], Dic3×C2×C10, D5×C22×C6, C23×D15, C22×C3⋊D20
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C3⋊D4 [×4], C22×S3 [×7], C22×D4, D20 [×4], C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, S3×D5, C2×D20 [×6], C23×D5, C22×C3⋊D4, C3⋊D20 [×4], C2×S3×D5 [×3], C22×D20, C2×C3⋊D20 [×6], C22×S3×D5, C22×C3⋊D20
Generators and relations
G = < a,b,c,d,e | a2=b2=c3=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
►On 240 points
Generators in S240
(1 212)(2 213)(3 214)(4 215)(5 216)(6 217)(7 218)(8 219)(9 220)(10 201)(11 202)(12 203)(13 204)(14 205)(15 206)(16 207)(17 208)(18 209)(19 210)(20 211)(21 231)(22 232)(23 233)(24 234)(25 235)(26 236)(27 237)(28 238)(29 239)(30 240)(31 221)(32 222)(33 223)(34 224)(35 225)(36 226)(37 227)(38 228)(39 229)(40 230)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 159)(62 160)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 181)(89 182)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(97 190)(98 191)(99 192)(100 193)(101 169)(102 170)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)
(1 125)(2 126)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 136)(13 137)(14 138)(15 139)(16 140)(17 121)(18 122)(19 123)(20 124)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 217)(42 218)(43 219)(44 220)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 211)(56 212)(57 213)(58 214)(59 215)(60 216)(81 118)(82 119)(83 120)(84 101)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(141 240)(142 221)(143 222)(144 223)(145 224)(146 225)(147 226)(148 227)(149 228)(150 229)(151 230)(152 231)(153 232)(154 233)(155 234)(156 235)(157 236)(158 237)(159 238)(160 239)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)
(1 105 36)(2 37 106)(3 107 38)(4 39 108)(5 109 40)(6 21 110)(7 111 22)(8 23 112)(9 113 24)(10 25 114)(11 115 26)(12 27 116)(13 117 28)(14 29 118)(15 119 30)(16 31 120)(17 101 32)(18 33 102)(19 103 34)(20 35 104)(41 152 186)(42 187 153)(43 154 188)(44 189 155)(45 156 190)(46 191 157)(47 158 192)(48 193 159)(49 160 194)(50 195 141)(51 142 196)(52 197 143)(53 144 198)(54 199 145)(55 146 200)(56 181 147)(57 148 182)(58 183 149)(59 150 184)(60 185 151)(61 137 100)(62 81 138)(63 139 82)(64 83 140)(65 121 84)(66 85 122)(67 123 86)(68 87 124)(69 125 88)(70 89 126)(71 127 90)(72 91 128)(73 129 92)(74 93 130)(75 131 94)(76 95 132)(77 133 96)(78 97 134)(79 135 98)(80 99 136)(161 234 220)(162 201 235)(163 236 202)(164 203 237)(165 238 204)(166 205 239)(167 240 206)(168 207 221)(169 222 208)(170 209 223)(171 224 210)(172 211 225)(173 226 212)(174 213 227)(175 228 214)(176 215 229)(177 230 216)(178 217 231)(179 232 218)(180 219 233)
(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)
(1 201)(2 220)(3 219)(4 218)(5 217)(6 216)(7 215)(8 214)(9 213)(10 212)(11 211)(12 210)(13 209)(14 208)(15 207)(16 206)(17 205)(18 204)(19 203)(20 202)(21 230)(22 229)(23 228)(24 227)(25 226)(26 225)(27 224)(28 223)(29 222)(30 221)(31 240)(32 239)(33 238)(34 237)(35 236)(36 235)(37 234)(38 233)(39 232)(40 231)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(61 144)(62 143)(63 142)(64 141)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 197)(82 196)(83 195)(84 194)(85 193)(86 192)(87 191)(88 190)(89 189)(90 188)(91 187)(92 186)(93 185)(94 184)(95 183)(96 182)(97 181)(98 200)(99 199)(100 198)(101 166)(102 165)(103 164)(104 163)(105 162)(106 161)(107 180)(108 179)(109 178)(110 177)(111 176)(112 175)(113 174)(114 173)(115 172)(116 171)(117 170)(118 169)(119 168)(120 167)
G:=sub<Sym(240)| (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,211)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,221)(32,222)(33,223)(34,224)(35,225)(36,226)(37,227)(38,228)(39,229)(40,230)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168), (1,125)(2,126)(3,127)(4,128)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,121)(18,122)(19,123)(20,124)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,217)(42,218)(43,219)(44,220)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,211)(56,212)(57,213)(58,214)(59,215)(60,216)(81,118)(82,119)(83,120)(84,101)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(141,240)(142,221)(143,222)(144,223)(145,224)(146,225)(147,226)(148,227)(149,228)(150,229)(151,230)(152,231)(153,232)(154,233)(155,234)(156,235)(157,236)(158,237)(159,238)(160,239)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (1,105,36)(2,37,106)(3,107,38)(4,39,108)(5,109,40)(6,21,110)(7,111,22)(8,23,112)(9,113,24)(10,25,114)(11,115,26)(12,27,116)(13,117,28)(14,29,118)(15,119,30)(16,31,120)(17,101,32)(18,33,102)(19,103,34)(20,35,104)(41,152,186)(42,187,153)(43,154,188)(44,189,155)(45,156,190)(46,191,157)(47,158,192)(48,193,159)(49,160,194)(50,195,141)(51,142,196)(52,197,143)(53,144,198)(54,199,145)(55,146,200)(56,181,147)(57,148,182)(58,183,149)(59,150,184)(60,185,151)(61,137,100)(62,81,138)(63,139,82)(64,83,140)(65,121,84)(66,85,122)(67,123,86)(68,87,124)(69,125,88)(70,89,126)(71,127,90)(72,91,128)(73,129,92)(74,93,130)(75,131,94)(76,95,132)(77,133,96)(78,97,134)(79,135,98)(80,99,136)(161,234,220)(162,201,235)(163,236,202)(164,203,237)(165,238,204)(166,205,239)(167,240,206)(168,207,221)(169,222,208)(170,209,223)(171,224,210)(172,211,225)(173,226,212)(174,213,227)(175,228,214)(176,215,229)(177,230,216)(178,217,231)(179,232,218)(180,219,233), (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), (1,201)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,230)(22,229)(23,228)(24,227)(25,226)(26,225)(27,224)(28,223)(29,222)(30,221)(31,240)(32,239)(33,238)(34,237)(35,236)(36,235)(37,234)(38,233)(39,232)(40,231)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,144)(62,143)(63,142)(64,141)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,197)(82,196)(83,195)(84,194)(85,193)(86,192)(87,191)(88,190)(89,189)(90,188)(91,187)(92,186)(93,185)(94,184)(95,183)(96,182)(97,181)(98,200)(99,199)(100,198)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,180)(108,179)(109,178)(110,177)(111,176)(112,175)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,167)>;
G:=Group( (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,211)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,221)(32,222)(33,223)(34,224)(35,225)(36,226)(37,227)(38,228)(39,229)(40,230)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168), (1,125)(2,126)(3,127)(4,128)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,121)(18,122)(19,123)(20,124)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,217)(42,218)(43,219)(44,220)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,211)(56,212)(57,213)(58,214)(59,215)(60,216)(81,118)(82,119)(83,120)(84,101)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(141,240)(142,221)(143,222)(144,223)(145,224)(146,225)(147,226)(148,227)(149,228)(150,229)(151,230)(152,231)(153,232)(154,233)(155,234)(156,235)(157,236)(158,237)(159,238)(160,239)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (1,105,36)(2,37,106)(3,107,38)(4,39,108)(5,109,40)(6,21,110)(7,111,22)(8,23,112)(9,113,24)(10,25,114)(11,115,26)(12,27,116)(13,117,28)(14,29,118)(15,119,30)(16,31,120)(17,101,32)(18,33,102)(19,103,34)(20,35,104)(41,152,186)(42,187,153)(43,154,188)(44,189,155)(45,156,190)(46,191,157)(47,158,192)(48,193,159)(49,160,194)(50,195,141)(51,142,196)(52,197,143)(53,144,198)(54,199,145)(55,146,200)(56,181,147)(57,148,182)(58,183,149)(59,150,184)(60,185,151)(61,137,100)(62,81,138)(63,139,82)(64,83,140)(65,121,84)(66,85,122)(67,123,86)(68,87,124)(69,125,88)(70,89,126)(71,127,90)(72,91,128)(73,129,92)(74,93,130)(75,131,94)(76,95,132)(77,133,96)(78,97,134)(79,135,98)(80,99,136)(161,234,220)(162,201,235)(163,236,202)(164,203,237)(165,238,204)(166,205,239)(167,240,206)(168,207,221)(169,222,208)(170,209,223)(171,224,210)(172,211,225)(173,226,212)(174,213,227)(175,228,214)(176,215,229)(177,230,216)(178,217,231)(179,232,218)(180,219,233), (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), (1,201)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,230)(22,229)(23,228)(24,227)(25,226)(26,225)(27,224)(28,223)(29,222)(30,221)(31,240)(32,239)(33,238)(34,237)(35,236)(36,235)(37,234)(38,233)(39,232)(40,231)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,144)(62,143)(63,142)(64,141)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,197)(82,196)(83,195)(84,194)(85,193)(86,192)(87,191)(88,190)(89,189)(90,188)(91,187)(92,186)(93,185)(94,184)(95,183)(96,182)(97,181)(98,200)(99,199)(100,198)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,180)(108,179)(109,178)(110,177)(111,176)(112,175)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,167) );
G=PermutationGroup([(1,212),(2,213),(3,214),(4,215),(5,216),(6,217),(7,218),(8,219),(9,220),(10,201),(11,202),(12,203),(13,204),(14,205),(15,206),(16,207),(17,208),(18,209),(19,210),(20,211),(21,231),(22,232),(23,233),(24,234),(25,235),(26,236),(27,237),(28,238),(29,239),(30,240),(31,221),(32,222),(33,223),(34,224),(35,225),(36,226),(37,227),(38,228),(39,229),(40,230),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,121),(53,122),(54,123),(55,124),(56,125),(57,126),(58,127),(59,128),(60,129),(61,159),(62,160),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,194),(82,195),(83,196),(84,197),(85,198),(86,199),(87,200),(88,181),(89,182),(90,183),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(97,190),(98,191),(99,192),(100,193),(101,169),(102,170),(103,171),(104,172),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168)], [(1,125),(2,126),(3,127),(4,128),(5,129),(6,130),(7,131),(8,132),(9,133),(10,134),(11,135),(12,136),(13,137),(14,138),(15,139),(16,140),(17,121),(18,122),(19,123),(20,124),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,217),(42,218),(43,219),(44,220),(45,201),(46,202),(47,203),(48,204),(49,205),(50,206),(51,207),(52,208),(53,209),(54,210),(55,211),(56,212),(57,213),(58,214),(59,215),(60,216),(81,118),(82,119),(83,120),(84,101),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(141,240),(142,221),(143,222),(144,223),(145,224),(146,225),(147,226),(148,227),(149,228),(150,229),(151,230),(152,231),(153,232),(154,233),(155,234),(156,235),(157,236),(158,237),(159,238),(160,239),(161,189),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196),(169,197),(170,198),(171,199),(172,200),(173,181),(174,182),(175,183),(176,184),(177,185),(178,186),(179,187),(180,188)], [(1,105,36),(2,37,106),(3,107,38),(4,39,108),(5,109,40),(6,21,110),(7,111,22),(8,23,112),(9,113,24),(10,25,114),(11,115,26),(12,27,116),(13,117,28),(14,29,118),(15,119,30),(16,31,120),(17,101,32),(18,33,102),(19,103,34),(20,35,104),(41,152,186),(42,187,153),(43,154,188),(44,189,155),(45,156,190),(46,191,157),(47,158,192),(48,193,159),(49,160,194),(50,195,141),(51,142,196),(52,197,143),(53,144,198),(54,199,145),(55,146,200),(56,181,147),(57,148,182),(58,183,149),(59,150,184),(60,185,151),(61,137,100),(62,81,138),(63,139,82),(64,83,140),(65,121,84),(66,85,122),(67,123,86),(68,87,124),(69,125,88),(70,89,126),(71,127,90),(72,91,128),(73,129,92),(74,93,130),(75,131,94),(76,95,132),(77,133,96),(78,97,134),(79,135,98),(80,99,136),(161,234,220),(162,201,235),(163,236,202),(164,203,237),(165,238,204),(166,205,239),(167,240,206),(168,207,221),(169,222,208),(170,209,223),(171,224,210),(172,211,225),(173,226,212),(174,213,227),(175,228,214),(176,215,229),(177,230,216),(178,217,231),(179,232,218),(180,219,233)], [(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)], [(1,201),(2,220),(3,219),(4,218),(5,217),(6,216),(7,215),(8,214),(9,213),(10,212),(11,211),(12,210),(13,209),(14,208),(15,207),(16,206),(17,205),(18,204),(19,203),(20,202),(21,230),(22,229),(23,228),(24,227),(25,226),(26,225),(27,224),(28,223),(29,222),(30,221),(31,240),(32,239),(33,238),(34,237),(35,236),(36,235),(37,234),(38,233),(39,232),(40,231),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,133),(58,132),(59,131),(60,130),(61,144),(62,143),(63,142),(64,141),(65,160),(66,159),(67,158),(68,157),(69,156),(70,155),(71,154),(72,153),(73,152),(74,151),(75,150),(76,149),(77,148),(78,147),(79,146),(80,145),(81,197),(82,196),(83,195),(84,194),(85,193),(86,192),(87,191),(88,190),(89,189),(90,188),(91,187),(92,186),(93,185),(94,184),(95,183),(96,182),(97,181),(98,200),(99,199),(100,198),(101,166),(102,165),(103,164),(104,163),(105,162),(106,161),(107,180),(108,179),(109,178),(110,177),(111,176),(112,175),(113,174),(114,173),(115,172),(116,171),(117,170),(118,169),(119,168),(120,167)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 44 | 60 | 0 | 0 |
| 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,47,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,60,17,0,0,0,0,0,0,60,0,0,0,1,0],[1,0,0,0,0,0,44,44,0,0,0,60,17,0,0,0,0,0,0,1,0,0,0,1,0] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10N | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | D20 | S3×D5 | C3⋊D20 | C2×S3×D5 |
| kernel | C22×C3⋊D20 | C2×C3⋊D20 | Dic3×C2×C10 | D5×C22×C6 | C23×D15 | C23×D5 | C2×C30 | C22×Dic3 | C22×D5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 1 | 12 | 2 | 8 | 16 | 2 | 8 | 6 |
In GAP, Magma, Sage, TeX
C_2^2\times C_3\rtimes D_{20} % in TeX
G:=Group("C2^2xC3:D20"); // GroupNames label
G:=SmallGroup(480,1119);
// by ID
G=gap.SmallGroup(480,1119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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