direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C3⋊D20, D30⋊9C23, C30.47C24, (C2×C6)⋊9D20, C30⋊5(C2×D4), C6⋊3(C2×D20), (C2×C30)⋊14D4, C15⋊6(C22×D4), C3⋊3(C22×D20), (C6×D5)⋊7C23, (C23×D5)⋊7S3, D10⋊7(C22×S3), (C22×D5)⋊15D6, (C23×D15)⋊8C2, C23.71(S3×D5), C6.47(C23×D5), (C2×Dic3)⋊24D10, C10.47(S3×C23), Dic3⋊5(C22×D5), (C22×Dic3)⋊9D5, (C5×Dic3)⋊8C23, (C2×C30).250C23, (C22×C10).119D6, (C22×C6).102D10, (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
Generators and relations for C22×C3⋊D20
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 >
Subgroups: 2684 in 472 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C20, D10, D10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C22×D4, D20, C2×C20, C22×D5, C22×D5, C22×C10, C22×Dic3, C2×C3⋊D4, S3×C23, C23×C6, C5×Dic3, C6×D5, C6×D5, D30, D30, C2×C30, C2×D20, C22×C20, C23×D5, C23×D5, C22×C3⋊D4, C3⋊D20, C10×Dic3, D5×C2×C6, D5×C2×C6, C22×D15, C22×D15, C22×C30, C22×D20, C2×C3⋊D20, Dic3×C2×C10, D5×C22×C6, C23×D15, C22×C3⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C3⋊D4, C22×S3, C22×D4, D20, C22×D5, C2×C3⋊D4, S3×C23, S3×D5, C2×D20, C23×D5, C22×C3⋊D4, C3⋊D20, C2×S3×D5, C22×D20, C2×C3⋊D20, C22×S3×D5, C22×C3⋊D20
►On 240 points
Generators in S240
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 98)(22 99)(23 100)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)(101 199)(102 200)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)(118 196)(119 197)(120 198)(161 233)(162 234)(163 235)(164 236)(165 237)(166 238)(167 239)(168 240)(169 221)(170 222)(171 223)(172 224)(173 225)(174 226)(175 227)(176 228)(177 229)(178 230)(179 231)(180 232)
(1 137)(2 138)(3 139)(4 140)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 155)(22 156)(23 157)(24 158)(25 159)(26 160)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 206)(42 207)(43 208)(44 209)(45 210)(46 211)(47 212)(48 213)(49 214)(50 215)(51 216)(52 217)(53 218)(54 219)(55 220)(56 201)(57 202)(58 203)(59 204)(60 205)(61 96)(62 97)(63 98)(64 99)(65 100)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(101 239)(102 240)(103 221)(104 222)(105 223)(106 224)(107 225)(108 226)(109 227)(110 228)(111 229)(112 230)(113 231)(114 232)(115 233)(116 234)(117 235)(118 236)(119 237)(120 238)(161 193)(162 194)(163 195)(164 196)(165 197)(166 198)(167 199)(168 200)(169 181)(170 182)(171 183)(172 184)(173 185)(174 186)(175 187)(176 188)(177 189)(178 190)(179 191)(180 192)
(1 99 224)(2 225 100)(3 81 226)(4 227 82)(5 83 228)(6 229 84)(7 85 230)(8 231 86)(9 87 232)(10 233 88)(11 89 234)(12 235 90)(13 91 236)(14 237 92)(15 93 238)(16 239 94)(17 95 240)(18 221 96)(19 97 222)(20 223 98)(21 213 171)(22 172 214)(23 215 173)(24 174 216)(25 217 175)(26 176 218)(27 219 177)(28 178 220)(29 201 179)(30 180 202)(31 203 161)(32 162 204)(33 205 163)(34 164 206)(35 207 165)(36 166 208)(37 209 167)(38 168 210)(39 211 169)(40 170 212)(41 148 196)(42 197 149)(43 150 198)(44 199 151)(45 152 200)(46 181 153)(47 154 182)(48 183 155)(49 156 184)(50 185 157)(51 158 186)(52 187 159)(53 160 188)(54 189 141)(55 142 190)(56 191 143)(57 144 192)(58 193 145)(59 146 194)(60 195 147)(61 134 103)(62 104 135)(63 136 105)(64 106 137)(65 138 107)(66 108 139)(67 140 109)(68 110 121)(69 122 111)(70 112 123)(71 124 113)(72 114 125)(73 126 115)(74 116 127)(75 128 117)(76 118 129)(77 130 119)(78 120 131)(79 132 101)(80 102 133)
(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)
(1 203)(2 202)(3 201)(4 220)(5 219)(6 218)(7 217)(8 216)(9 215)(10 214)(11 213)(12 212)(13 211)(14 210)(15 209)(16 208)(17 207)(18 206)(19 205)(20 204)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 134)(42 133)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 140)(56 139)(57 138)(58 137)(59 136)(60 135)(61 148)(62 147)(63 146)(64 145)(65 144)(66 143)(67 142)(68 141)(69 160)(70 159)(71 158)(72 157)(73 156)(74 155)(75 154)(76 153)(77 152)(78 151)(79 150)(80 149)(101 198)(102 197)(103 196)(104 195)(105 194)(106 193)(107 192)(108 191)(109 190)(110 189)(111 188)(112 187)(113 186)(114 185)(115 184)(116 183)(117 182)(118 181)(119 200)(120 199)(161 224)(162 223)(163 222)(164 221)(165 240)(166 239)(167 238)(168 237)(169 236)(170 235)(171 234)(172 233)(173 232)(174 231)(175 230)(176 229)(177 228)(178 227)(179 226)(180 225)
G:=sub<Sym(240)| (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,98)(22,99)(23,100)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(101,199)(102,200)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)(168,240)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,155)(22,156)(23,157)(24,158)(25,159)(26,160)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,201)(57,202)(58,203)(59,204)(60,205)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(101,239)(102,240)(103,221)(104,222)(105,223)(106,224)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,181)(170,182)(171,183)(172,184)(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192), (1,99,224)(2,225,100)(3,81,226)(4,227,82)(5,83,228)(6,229,84)(7,85,230)(8,231,86)(9,87,232)(10,233,88)(11,89,234)(12,235,90)(13,91,236)(14,237,92)(15,93,238)(16,239,94)(17,95,240)(18,221,96)(19,97,222)(20,223,98)(21,213,171)(22,172,214)(23,215,173)(24,174,216)(25,217,175)(26,176,218)(27,219,177)(28,178,220)(29,201,179)(30,180,202)(31,203,161)(32,162,204)(33,205,163)(34,164,206)(35,207,165)(36,166,208)(37,209,167)(38,168,210)(39,211,169)(40,170,212)(41,148,196)(42,197,149)(43,150,198)(44,199,151)(45,152,200)(46,181,153)(47,154,182)(48,183,155)(49,156,184)(50,185,157)(51,158,186)(52,187,159)(53,160,188)(54,189,141)(55,142,190)(56,191,143)(57,144,192)(58,193,145)(59,146,194)(60,195,147)(61,134,103)(62,104,135)(63,136,105)(64,106,137)(65,138,107)(66,108,139)(67,140,109)(68,110,121)(69,122,111)(70,112,123)(71,124,113)(72,114,125)(73,126,115)(74,116,127)(75,128,117)(76,118,129)(77,130,119)(78,120,131)(79,132,101)(80,102,133), (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), (1,203)(2,202)(3,201)(4,220)(5,219)(6,218)(7,217)(8,216)(9,215)(10,214)(11,213)(12,212)(13,211)(14,210)(15,209)(16,208)(17,207)(18,206)(19,205)(20,204)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,186)(114,185)(115,184)(116,183)(117,182)(118,181)(119,200)(120,199)(161,224)(162,223)(163,222)(164,221)(165,240)(166,239)(167,238)(168,237)(169,236)(170,235)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)>;
G:=Group( (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,98)(22,99)(23,100)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(101,199)(102,200)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)(168,240)(169,221)(170,222)(171,223)(172,224)(173,225)(174,226)(175,227)(176,228)(177,229)(178,230)(179,231)(180,232), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,155)(22,156)(23,157)(24,158)(25,159)(26,160)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,201)(57,202)(58,203)(59,204)(60,205)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(101,239)(102,240)(103,221)(104,222)(105,223)(106,224)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,181)(170,182)(171,183)(172,184)(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192), (1,99,224)(2,225,100)(3,81,226)(4,227,82)(5,83,228)(6,229,84)(7,85,230)(8,231,86)(9,87,232)(10,233,88)(11,89,234)(12,235,90)(13,91,236)(14,237,92)(15,93,238)(16,239,94)(17,95,240)(18,221,96)(19,97,222)(20,223,98)(21,213,171)(22,172,214)(23,215,173)(24,174,216)(25,217,175)(26,176,218)(27,219,177)(28,178,220)(29,201,179)(30,180,202)(31,203,161)(32,162,204)(33,205,163)(34,164,206)(35,207,165)(36,166,208)(37,209,167)(38,168,210)(39,211,169)(40,170,212)(41,148,196)(42,197,149)(43,150,198)(44,199,151)(45,152,200)(46,181,153)(47,154,182)(48,183,155)(49,156,184)(50,185,157)(51,158,186)(52,187,159)(53,160,188)(54,189,141)(55,142,190)(56,191,143)(57,144,192)(58,193,145)(59,146,194)(60,195,147)(61,134,103)(62,104,135)(63,136,105)(64,106,137)(65,138,107)(66,108,139)(67,140,109)(68,110,121)(69,122,111)(70,112,123)(71,124,113)(72,114,125)(73,126,115)(74,116,127)(75,128,117)(76,118,129)(77,130,119)(78,120,131)(79,132,101)(80,102,133), (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), (1,203)(2,202)(3,201)(4,220)(5,219)(6,218)(7,217)(8,216)(9,215)(10,214)(11,213)(12,212)(13,211)(14,210)(15,209)(16,208)(17,207)(18,206)(19,205)(20,204)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,82)(29,81)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,186)(114,185)(115,184)(116,183)(117,182)(118,181)(119,200)(120,199)(161,224)(162,223)(163,222)(164,221)(165,240)(166,239)(167,238)(168,237)(169,236)(170,235)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225) );
G=PermutationGroup([[(1,214),(2,215),(3,216),(4,217),(5,218),(6,219),(7,220),(8,201),(9,202),(10,203),(11,204),(12,205),(13,206),(14,207),(15,208),(16,209),(17,210),(18,211),(19,212),(20,213),(21,98),(22,99),(23,100),(24,81),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,141),(70,142),(71,143),(72,144),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152),(101,199),(102,200),(103,181),(104,182),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(113,191),(114,192),(115,193),(116,194),(117,195),(118,196),(119,197),(120,198),(161,233),(162,234),(163,235),(164,236),(165,237),(166,238),(167,239),(168,240),(169,221),(170,222),(171,223),(172,224),(173,225),(174,226),(175,227),(176,228),(177,229),(178,230),(179,231),(180,232)], [(1,137),(2,138),(3,139),(4,140),(5,121),(6,122),(7,123),(8,124),(9,125),(10,126),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,155),(22,156),(23,157),(24,158),(25,159),(26,160),(27,141),(28,142),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,206),(42,207),(43,208),(44,209),(45,210),(46,211),(47,212),(48,213),(49,214),(50,215),(51,216),(52,217),(53,218),(54,219),(55,220),(56,201),(57,202),(58,203),(59,204),(60,205),(61,96),(62,97),(63,98),(64,99),(65,100),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(101,239),(102,240),(103,221),(104,222),(105,223),(106,224),(107,225),(108,226),(109,227),(110,228),(111,229),(112,230),(113,231),(114,232),(115,233),(116,234),(117,235),(118,236),(119,237),(120,238),(161,193),(162,194),(163,195),(164,196),(165,197),(166,198),(167,199),(168,200),(169,181),(170,182),(171,183),(172,184),(173,185),(174,186),(175,187),(176,188),(177,189),(178,190),(179,191),(180,192)], [(1,99,224),(2,225,100),(3,81,226),(4,227,82),(5,83,228),(6,229,84),(7,85,230),(8,231,86),(9,87,232),(10,233,88),(11,89,234),(12,235,90),(13,91,236),(14,237,92),(15,93,238),(16,239,94),(17,95,240),(18,221,96),(19,97,222),(20,223,98),(21,213,171),(22,172,214),(23,215,173),(24,174,216),(25,217,175),(26,176,218),(27,219,177),(28,178,220),(29,201,179),(30,180,202),(31,203,161),(32,162,204),(33,205,163),(34,164,206),(35,207,165),(36,166,208),(37,209,167),(38,168,210),(39,211,169),(40,170,212),(41,148,196),(42,197,149),(43,150,198),(44,199,151),(45,152,200),(46,181,153),(47,154,182),(48,183,155),(49,156,184),(50,185,157),(51,158,186),(52,187,159),(53,160,188),(54,189,141),(55,142,190),(56,191,143),(57,144,192),(58,193,145),(59,146,194),(60,195,147),(61,134,103),(62,104,135),(63,136,105),(64,106,137),(65,138,107),(66,108,139),(67,140,109),(68,110,121),(69,122,111),(70,112,123),(71,124,113),(72,114,125),(73,126,115),(74,116,127),(75,128,117),(76,118,129),(77,130,119),(78,120,131),(79,132,101),(80,102,133)], [(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)], [(1,203),(2,202),(3,201),(4,220),(5,219),(6,218),(7,217),(8,216),(9,215),(10,214),(11,213),(12,212),(13,211),(14,210),(15,209),(16,208),(17,207),(18,206),(19,205),(20,204),(21,89),(22,88),(23,87),(24,86),(25,85),(26,84),(27,83),(28,82),(29,81),(30,100),(31,99),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,134),(42,133),(43,132),(44,131),(45,130),(46,129),(47,128),(48,127),(49,126),(50,125),(51,124),(52,123),(53,122),(54,121),(55,140),(56,139),(57,138),(58,137),(59,136),(60,135),(61,148),(62,147),(63,146),(64,145),(65,144),(66,143),(67,142),(68,141),(69,160),(70,159),(71,158),(72,157),(73,156),(74,155),(75,154),(76,153),(77,152),(78,151),(79,150),(80,149),(101,198),(102,197),(103,196),(104,195),(105,194),(106,193),(107,192),(108,191),(109,190),(110,189),(111,188),(112,187),(113,186),(114,185),(115,184),(116,183),(117,182),(118,181),(119,200),(120,199),(161,224),(162,223),(163,222),(164,221),(165,240),(166,239),(167,238),(168,237),(169,236),(170,235),(171,234),(172,233),(173,232),(174,231),(175,230),(176,229),(177,228),(178,227),(179,226),(180,225)]])
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 |
Matrix representation of C22×C3⋊D20 ►in 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] >;
C22×C3⋊D20 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