direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C10×C3⋊D4, C30.98C24, C6⋊3(D4×C10), (C2×C30)⋊35D4, C30⋊18(C2×D4), C24⋊6(C5×S3), C23⋊5(S3×C10), (C23×C6)⋊6C10, (C23×C10)⋊8S3, C15⋊19(C22×D4), (S3×C23)⋊5C10, (C2×C30)⋊14C23, (C23×C30)⋊10C2, D6⋊3(C22×C10), (C22×C10)⋊16D6, (S3×C10)⋊12C23, C6.15(C23×C10), C10.83(S3×C23), (C22×C30)⋊23C22, (C5×Dic3)⋊10C23, (C22×Dic3)⋊9C10, Dic3⋊2(C22×C10), (C10×Dic3)⋊38C22, C3⋊3(D4×C2×C10), (C2×C6)⋊9(C5×D4), C22⋊3(S3×C2×C10), (S3×C22×C10)⋊11C2, (S3×C2×C10)⋊23C22, (C22×C6)⋊7(C2×C10), (C2×C6)⋊3(C22×C10), (Dic3×C2×C10)⋊20C2, C2.15(S3×C22×C10), (C22×S3)⋊7(C2×C10), (C2×C10)⋊11(C22×S3), (C2×Dic3)⋊11(C2×C10), SmallGroup(480,1164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C10×C3⋊D4
G = < a,b,c,d,e | a2=b10=c3=d4=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: 996 in 472 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, C23, C10, C10, C10, Dic3, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C24, C20, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C5×S3, C30, C30, C30, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C22×Dic3, C2×C3⋊D4, S3×C23, C23×C6, C5×Dic3, S3×C10, S3×C10, C2×C30, C2×C30, C22×C20, D4×C10, C23×C10, C23×C10, C22×C3⋊D4, C10×Dic3, C5×C3⋊D4, S3×C2×C10, S3×C2×C10, C22×C30, C22×C30, C22×C30, D4×C2×C10, Dic3×C2×C10, C10×C3⋊D4, S3×C22×C10, C23×C30, C2×C10×C3⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C24, C2×C10, C3⋊D4, C22×S3, C5×S3, C22×D4, C5×D4, C22×C10, C2×C3⋊D4, S3×C23, S3×C10, D4×C10, C23×C10, C22×C3⋊D4, C5×C3⋊D4, S3×C2×C10, D4×C2×C10, C10×C3⋊D4, S3×C22×C10, C2×C10×C3⋊D4
►On 240 points
Generators in S240
(1 202)(2 203)(3 204)(4 205)(5 206)(6 207)(7 208)(8 209)(9 210)(10 201)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 111)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 101)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 71)(40 72)(41 89)(42 90)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(91 239)(92 240)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(131 166)(132 167)(133 168)(134 169)(135 170)(136 161)(137 162)(138 163)(139 164)(140 165)(141 176)(142 177)(143 178)(144 179)(145 180)(146 171)(147 172)(148 173)(149 174)(150 175)(181 216)(182 217)(183 218)(184 219)(185 220)(186 211)(187 212)(188 213)(189 214)(190 215)(191 226)(192 227)(193 228)(194 229)(195 230)(196 221)(197 222)(198 223)(199 224)(200 225)
(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 41 189)(2 42 190)(3 43 181)(4 44 182)(5 45 183)(6 46 184)(7 47 185)(8 48 186)(9 49 187)(10 50 188)(11 158 180)(12 159 171)(13 160 172)(14 151 173)(15 152 174)(16 153 175)(17 154 176)(18 155 177)(19 156 178)(20 157 179)(21 240 164)(22 231 165)(23 232 166)(24 233 167)(25 234 168)(26 235 169)(27 236 170)(28 237 161)(29 238 162)(30 239 163)(31 53 193)(32 54 194)(33 55 195)(34 56 196)(35 57 197)(36 58 198)(37 59 199)(38 60 200)(39 51 191)(40 52 192)(61 228 73)(62 229 74)(63 230 75)(64 221 76)(65 222 77)(66 223 78)(67 224 79)(68 225 80)(69 226 71)(70 227 72)(81 216 204)(82 217 205)(83 218 206)(84 219 207)(85 220 208)(86 211 209)(87 212 210)(88 213 201)(89 214 202)(90 215 203)(91 138 101)(92 139 102)(93 140 103)(94 131 104)(95 132 105)(96 133 106)(97 134 107)(98 135 108)(99 136 109)(100 137 110)(111 122 144)(112 123 145)(113 124 146)(114 125 147)(115 126 148)(116 127 149)(117 128 150)(118 129 141)(119 130 142)(120 121 143)
(1 144 99 59)(2 145 100 60)(3 146 91 51)(4 147 92 52)(5 148 93 53)(6 149 94 54)(7 150 95 55)(8 141 96 56)(9 142 97 57)(10 143 98 58)(11 29 225 215)(12 30 226 216)(13 21 227 217)(14 22 228 218)(15 23 229 219)(16 24 230 220)(17 25 221 211)(18 26 222 212)(19 27 223 213)(20 28 224 214)(31 45 126 140)(32 46 127 131)(33 47 128 132)(34 48 129 133)(35 49 130 134)(36 50 121 135)(37 41 122 136)(38 42 123 137)(39 43 124 138)(40 44 125 139)(61 206 173 231)(62 207 174 232)(63 208 175 233)(64 209 176 234)(65 210 177 235)(66 201 178 236)(67 202 179 237)(68 203 180 238)(69 204 171 239)(70 205 172 240)(71 81 159 163)(72 82 160 164)(73 83 151 165)(74 84 152 166)(75 85 153 167)(76 86 154 168)(77 87 155 169)(78 88 156 170)(79 89 157 161)(80 90 158 162)(101 191 181 113)(102 192 182 114)(103 193 183 115)(104 194 184 116)(105 195 185 117)(106 196 186 118)(107 197 187 119)(108 198 188 120)(109 199 189 111)(110 200 190 112)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 151)(20 152)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 198)(32 199)(33 200)(34 191)(35 192)(36 193)(37 194)(38 195)(39 196)(40 197)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 101)(49 102)(50 103)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(131 189)(132 190)(133 181)(134 182)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 146)(142 147)(143 148)(144 149)(145 150)(161 219)(162 220)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)(169 217)(170 218)(171 176)(172 177)(173 178)(174 179)(175 180)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
G:=sub<Sym(240)| (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,111)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,101)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72)(41,89)(42,90)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(91,239)(92,240)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,166)(132,167)(133,168)(134,169)(135,170)(136,161)(137,162)(138,163)(139,164)(140,165)(141,176)(142,177)(143,178)(144,179)(145,180)(146,171)(147,172)(148,173)(149,174)(150,175)(181,216)(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215)(191,226)(192,227)(193,228)(194,229)(195,230)(196,221)(197,222)(198,223)(199,224)(200,225), (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), (1,41,189)(2,42,190)(3,43,181)(4,44,182)(5,45,183)(6,46,184)(7,47,185)(8,48,186)(9,49,187)(10,50,188)(11,158,180)(12,159,171)(13,160,172)(14,151,173)(15,152,174)(16,153,175)(17,154,176)(18,155,177)(19,156,178)(20,157,179)(21,240,164)(22,231,165)(23,232,166)(24,233,167)(25,234,168)(26,235,169)(27,236,170)(28,237,161)(29,238,162)(30,239,163)(31,53,193)(32,54,194)(33,55,195)(34,56,196)(35,57,197)(36,58,198)(37,59,199)(38,60,200)(39,51,191)(40,52,192)(61,228,73)(62,229,74)(63,230,75)(64,221,76)(65,222,77)(66,223,78)(67,224,79)(68,225,80)(69,226,71)(70,227,72)(81,216,204)(82,217,205)(83,218,206)(84,219,207)(85,220,208)(86,211,209)(87,212,210)(88,213,201)(89,214,202)(90,215,203)(91,138,101)(92,139,102)(93,140,103)(94,131,104)(95,132,105)(96,133,106)(97,134,107)(98,135,108)(99,136,109)(100,137,110)(111,122,144)(112,123,145)(113,124,146)(114,125,147)(115,126,148)(116,127,149)(117,128,150)(118,129,141)(119,130,142)(120,121,143), (1,144,99,59)(2,145,100,60)(3,146,91,51)(4,147,92,52)(5,148,93,53)(6,149,94,54)(7,150,95,55)(8,141,96,56)(9,142,97,57)(10,143,98,58)(11,29,225,215)(12,30,226,216)(13,21,227,217)(14,22,228,218)(15,23,229,219)(16,24,230,220)(17,25,221,211)(18,26,222,212)(19,27,223,213)(20,28,224,214)(31,45,126,140)(32,46,127,131)(33,47,128,132)(34,48,129,133)(35,49,130,134)(36,50,121,135)(37,41,122,136)(38,42,123,137)(39,43,124,138)(40,44,125,139)(61,206,173,231)(62,207,174,232)(63,208,175,233)(64,209,176,234)(65,210,177,235)(66,201,178,236)(67,202,179,237)(68,203,180,238)(69,204,171,239)(70,205,172,240)(71,81,159,163)(72,82,160,164)(73,83,151,165)(74,84,152,166)(75,85,153,167)(76,86,154,168)(77,87,155,169)(78,88,156,170)(79,89,157,161)(80,90,158,162)(101,191,181,113)(102,192,182,114)(103,193,183,115)(104,194,184,116)(105,195,185,117)(106,196,186,118)(107,197,187,119)(108,198,188,120)(109,199,189,111)(110,200,190,112), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,198)(32,199)(33,200)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(111,127)(112,128)(113,129)(114,130)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(131,189)(132,190)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,146)(142,147)(143,148)(144,149)(145,150)(161,219)(162,220)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216)(169,217)(170,218)(171,176)(172,177)(173,178)(174,179)(175,180)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240)>;
G:=Group( (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,209)(9,210)(10,201)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,111)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,101)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72)(41,89)(42,90)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(91,239)(92,240)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,166)(132,167)(133,168)(134,169)(135,170)(136,161)(137,162)(138,163)(139,164)(140,165)(141,176)(142,177)(143,178)(144,179)(145,180)(146,171)(147,172)(148,173)(149,174)(150,175)(181,216)(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215)(191,226)(192,227)(193,228)(194,229)(195,230)(196,221)(197,222)(198,223)(199,224)(200,225), (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), (1,41,189)(2,42,190)(3,43,181)(4,44,182)(5,45,183)(6,46,184)(7,47,185)(8,48,186)(9,49,187)(10,50,188)(11,158,180)(12,159,171)(13,160,172)(14,151,173)(15,152,174)(16,153,175)(17,154,176)(18,155,177)(19,156,178)(20,157,179)(21,240,164)(22,231,165)(23,232,166)(24,233,167)(25,234,168)(26,235,169)(27,236,170)(28,237,161)(29,238,162)(30,239,163)(31,53,193)(32,54,194)(33,55,195)(34,56,196)(35,57,197)(36,58,198)(37,59,199)(38,60,200)(39,51,191)(40,52,192)(61,228,73)(62,229,74)(63,230,75)(64,221,76)(65,222,77)(66,223,78)(67,224,79)(68,225,80)(69,226,71)(70,227,72)(81,216,204)(82,217,205)(83,218,206)(84,219,207)(85,220,208)(86,211,209)(87,212,210)(88,213,201)(89,214,202)(90,215,203)(91,138,101)(92,139,102)(93,140,103)(94,131,104)(95,132,105)(96,133,106)(97,134,107)(98,135,108)(99,136,109)(100,137,110)(111,122,144)(112,123,145)(113,124,146)(114,125,147)(115,126,148)(116,127,149)(117,128,150)(118,129,141)(119,130,142)(120,121,143), (1,144,99,59)(2,145,100,60)(3,146,91,51)(4,147,92,52)(5,148,93,53)(6,149,94,54)(7,150,95,55)(8,141,96,56)(9,142,97,57)(10,143,98,58)(11,29,225,215)(12,30,226,216)(13,21,227,217)(14,22,228,218)(15,23,229,219)(16,24,230,220)(17,25,221,211)(18,26,222,212)(19,27,223,213)(20,28,224,214)(31,45,126,140)(32,46,127,131)(33,47,128,132)(34,48,129,133)(35,49,130,134)(36,50,121,135)(37,41,122,136)(38,42,123,137)(39,43,124,138)(40,44,125,139)(61,206,173,231)(62,207,174,232)(63,208,175,233)(64,209,176,234)(65,210,177,235)(66,201,178,236)(67,202,179,237)(68,203,180,238)(69,204,171,239)(70,205,172,240)(71,81,159,163)(72,82,160,164)(73,83,151,165)(74,84,152,166)(75,85,153,167)(76,86,154,168)(77,87,155,169)(78,88,156,170)(79,89,157,161)(80,90,158,162)(101,191,181,113)(102,192,182,114)(103,193,183,115)(104,194,184,116)(105,195,185,117)(106,196,186,118)(107,197,187,119)(108,198,188,120)(109,199,189,111)(110,200,190,112), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,198)(32,199)(33,200)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(111,127)(112,128)(113,129)(114,130)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(131,189)(132,190)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,146)(142,147)(143,148)(144,149)(145,150)(161,219)(162,220)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216)(169,217)(170,218)(171,176)(172,177)(173,178)(174,179)(175,180)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240) );
G=PermutationGroup([[(1,202),(2,203),(3,204),(4,205),(5,206),(6,207),(7,208),(8,209),(9,210),(10,201),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,111),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,101),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,71),(40,72),(41,89),(42,90),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(91,239),(92,240),(93,231),(94,232),(95,233),(96,234),(97,235),(98,236),(99,237),(100,238),(121,156),(122,157),(123,158),(124,159),(125,160),(126,151),(127,152),(128,153),(129,154),(130,155),(131,166),(132,167),(133,168),(134,169),(135,170),(136,161),(137,162),(138,163),(139,164),(140,165),(141,176),(142,177),(143,178),(144,179),(145,180),(146,171),(147,172),(148,173),(149,174),(150,175),(181,216),(182,217),(183,218),(184,219),(185,220),(186,211),(187,212),(188,213),(189,214),(190,215),(191,226),(192,227),(193,228),(194,229),(195,230),(196,221),(197,222),(198,223),(199,224),(200,225)], [(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)], [(1,41,189),(2,42,190),(3,43,181),(4,44,182),(5,45,183),(6,46,184),(7,47,185),(8,48,186),(9,49,187),(10,50,188),(11,158,180),(12,159,171),(13,160,172),(14,151,173),(15,152,174),(16,153,175),(17,154,176),(18,155,177),(19,156,178),(20,157,179),(21,240,164),(22,231,165),(23,232,166),(24,233,167),(25,234,168),(26,235,169),(27,236,170),(28,237,161),(29,238,162),(30,239,163),(31,53,193),(32,54,194),(33,55,195),(34,56,196),(35,57,197),(36,58,198),(37,59,199),(38,60,200),(39,51,191),(40,52,192),(61,228,73),(62,229,74),(63,230,75),(64,221,76),(65,222,77),(66,223,78),(67,224,79),(68,225,80),(69,226,71),(70,227,72),(81,216,204),(82,217,205),(83,218,206),(84,219,207),(85,220,208),(86,211,209),(87,212,210),(88,213,201),(89,214,202),(90,215,203),(91,138,101),(92,139,102),(93,140,103),(94,131,104),(95,132,105),(96,133,106),(97,134,107),(98,135,108),(99,136,109),(100,137,110),(111,122,144),(112,123,145),(113,124,146),(114,125,147),(115,126,148),(116,127,149),(117,128,150),(118,129,141),(119,130,142),(120,121,143)], [(1,144,99,59),(2,145,100,60),(3,146,91,51),(4,147,92,52),(5,148,93,53),(6,149,94,54),(7,150,95,55),(8,141,96,56),(9,142,97,57),(10,143,98,58),(11,29,225,215),(12,30,226,216),(13,21,227,217),(14,22,228,218),(15,23,229,219),(16,24,230,220),(17,25,221,211),(18,26,222,212),(19,27,223,213),(20,28,224,214),(31,45,126,140),(32,46,127,131),(33,47,128,132),(34,48,129,133),(35,49,130,134),(36,50,121,135),(37,41,122,136),(38,42,123,137),(39,43,124,138),(40,44,125,139),(61,206,173,231),(62,207,174,232),(63,208,175,233),(64,209,176,234),(65,210,177,235),(66,201,178,236),(67,202,179,237),(68,203,180,238),(69,204,171,239),(70,205,172,240),(71,81,159,163),(72,82,160,164),(73,83,151,165),(74,84,152,166),(75,85,153,167),(76,86,154,168),(77,87,155,169),(78,88,156,170),(79,89,157,161),(80,90,158,162),(101,191,181,113),(102,192,182,114),(103,193,183,115),(104,194,184,116),(105,195,185,117),(106,196,186,118),(107,197,187,119),(108,198,188,120),(109,199,189,111),(110,200,190,112)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,91),(9,92),(10,93),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,151),(20,152),(21,87),(22,88),(23,89),(24,90),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,198),(32,199),(33,200),(34,191),(35,192),(36,193),(37,194),(38,195),(39,196),(40,197),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,101),(49,102),(50,103),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,227),(78,228),(79,229),(80,230),(111,127),(112,128),(113,129),(114,130),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126),(131,189),(132,190),(133,181),(134,182),(135,183),(136,184),(137,185),(138,186),(139,187),(140,188),(141,146),(142,147),(143,148),(144,149),(145,150),(161,219),(162,220),(163,211),(164,212),(165,213),(166,214),(167,215),(168,216),(169,217),(170,218),(171,176),(172,177),(173,178),(174,179),(175,180),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6O | 10A | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10BH | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×C3⋊D4 |
| kernel | C2×C10×C3⋊D4 | Dic3×C2×C10 | C10×C3⋊D4 | S3×C22×C10 | C23×C30 | C22×C3⋊D4 | C22×Dic3 | C2×C3⋊D4 | S3×C23 | C23×C6 | C23×C10 | C2×C30 | C22×C10 | C2×C10 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 4 | 4 | 48 | 4 | 4 | 1 | 4 | 7 | 8 | 4 | 16 | 28 | 32 |
Matrix representation of C2×C10×C3⋊D4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 20 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 20 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 43 | 9 |
| 0 | 0 | 0 | 52 | 18 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[20,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,20,0,0,0,0,0,20],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,43,52,0,0,0,9,18],[60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0] >;
C2×C10×C3⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{10}\times C_3\rtimes D_4 % in TeX
G:=Group("C2xC10xC3:D4"); // GroupNames label
G:=SmallGroup(480,1164);
// by ID
G=gap.SmallGroup(480,1164);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,2467,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^3=d^4=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