direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D4⋊S3, C30⋊9D8, C60.147D4, C60.227C23, C6⋊2(C5×D8), C3⋊3(C10×D8), C15⋊18(C2×D8), (C5×D4)⋊25D6, (C6×D4)⋊1C10, D4⋊3(S3×C10), (D4×C10)⋊8S3, (D4×C30)⋊15C2, (C2×D12)⋊8C10, D12⋊5(C2×C10), C6.44(D4×C10), C12.14(C5×D4), (C10×D12)⋊24C2, C30.427(C2×D4), (C2×C20).358D6, (C2×C30).181D4, (C5×D12)⋊35C22, (D4×C15)⋊34C22, C20.70(C3⋊D4), (C2×C60).357C22, C12.11(C22×C10), C20.200(C22×S3), (C2×C3⋊C8)⋊4C10, C3⋊C8⋊7(C2×C10), (C10×C3⋊C8)⋊18C2, (C2×D4)⋊1(C5×S3), C4.11(S3×C2×C10), C4.5(C5×C3⋊D4), (C3×D4)⋊3(C2×C10), (C5×C3⋊C8)⋊40C22, (C2×C6).38(C5×D4), C2.8(C10×C3⋊D4), (C2×C4).47(S3×C10), (C2×C12).30(C2×C10), C10.129(C2×C3⋊D4), C22.21(C5×C3⋊D4), (C2×C10).93(C3⋊D4), SmallGroup(480,810)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D4⋊S3
G = < a,b,c,d,e | a10=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Subgroups: 420 in 152 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C8, C2×C4, D4, D4, C23, C10, C10, C10, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C2×D4, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C5×S3, C30, C30, C30, C2×D8, C40, C2×C20, C5×D4, C5×D4, C22×C10, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C60, S3×C10, C2×C30, C2×C30, C2×C40, C5×D8, D4×C10, D4×C10, C2×D4⋊S3, C5×C3⋊C8, C5×D12, C5×D12, C2×C60, D4×C15, D4×C15, S3×C2×C10, C22×C30, C10×D8, C10×C3⋊C8, C5×D4⋊S3, C10×D12, D4×C30, C10×D4⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, D8, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C2×D8, C5×D4, C22×C10, D4⋊S3, C2×C3⋊D4, S3×C10, C5×D8, D4×C10, C2×D4⋊S3, C5×C3⋊D4, S3×C2×C10, C10×D8, C5×D4⋊S3, C10×C3⋊D4, C10×D4⋊S3
►On 240 points
Generators in S240
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190)(191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230)(231 232 233 234 235 236 237 238 239 240)
(1 223 158 195)(2 224 159 196)(3 225 160 197)(4 226 151 198)(5 227 152 199)(6 228 153 200)(7 229 154 191)(8 230 155 192)(9 221 156 193)(10 222 157 194)(11 66 60 86)(12 67 51 87)(13 68 52 88)(14 69 53 89)(15 70 54 90)(16 61 55 81)(17 62 56 82)(18 63 57 83)(19 64 58 84)(20 65 59 85)(21 120 185 135)(22 111 186 136)(23 112 187 137)(24 113 188 138)(25 114 189 139)(26 115 190 140)(27 116 181 131)(28 117 182 132)(29 118 183 133)(30 119 184 134)(31 143 206 78)(32 144 207 79)(33 145 208 80)(34 146 209 71)(35 147 210 72)(36 148 201 73)(37 149 202 74)(38 150 203 75)(39 141 204 76)(40 142 205 77)(41 127 102 214)(42 128 103 215)(43 129 104 216)(44 130 105 217)(45 121 106 218)(46 122 107 219)(47 123 108 220)(48 124 109 211)(49 125 110 212)(50 126 101 213)(91 171 166 233)(92 172 167 234)(93 173 168 235)(94 174 169 236)(95 175 170 237)(96 176 161 238)(97 177 162 239)(98 178 163 240)(99 179 164 231)(100 180 165 232)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(11 215)(12 216)(13 217)(14 218)(15 219)(16 220)(17 211)(18 212)(19 213)(20 214)(31 162)(32 163)(33 164)(34 165)(35 166)(36 167)(37 168)(38 169)(39 170)(40 161)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 81)(48 82)(49 83)(50 84)(51 129)(52 130)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 108)(62 109)(63 110)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 232)(72 233)(73 234)(74 235)(75 236)(76 237)(77 238)(78 239)(79 240)(80 231)(91 210)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 209)(111 192)(112 193)(113 194)(114 195)(115 196)(116 197)(117 198)(118 199)(119 200)(120 191)(131 225)(132 226)(133 227)(134 228)(135 229)(136 230)(137 221)(138 222)(139 223)(140 224)(141 175)(142 176)(143 177)(144 178)(145 179)(146 180)(147 171)(148 172)(149 173)(150 174)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 181)
(1 177 16)(2 178 17)(3 179 18)(4 180 19)(5 171 20)(6 172 11)(7 173 12)(8 174 13)(9 175 14)(10 176 15)(21 149 216)(22 150 217)(23 141 218)(24 142 219)(25 143 220)(26 144 211)(27 145 212)(28 146 213)(29 147 214)(30 148 215)(31 108 139)(32 109 140)(33 110 131)(34 101 132)(35 102 133)(36 103 134)(37 104 135)(38 105 136)(39 106 137)(40 107 138)(41 118 210)(42 119 201)(43 120 202)(44 111 203)(45 112 204)(46 113 205)(47 114 206)(48 115 207)(49 116 208)(50 117 209)(51 154 235)(52 155 236)(53 156 237)(54 157 238)(55 158 239)(56 159 240)(57 160 231)(58 151 232)(59 152 233)(60 153 234)(61 223 162)(62 224 163)(63 225 164)(64 226 165)(65 227 166)(66 228 167)(67 229 168)(68 230 169)(69 221 170)(70 222 161)(71 126 182)(72 127 183)(73 128 184)(74 129 185)(75 130 186)(76 121 187)(77 122 188)(78 123 189)(79 124 190)(80 125 181)(81 195 97)(82 196 98)(83 197 99)(84 198 100)(85 199 91)(86 200 92)(87 191 93)(88 192 94)(89 193 95)(90 194 96)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 177)(12 178)(13 179)(14 180)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 140)(22 131)(23 132)(24 133)(25 134)(26 135)(27 136)(28 137)(29 138)(30 139)(31 215)(32 216)(33 217)(34 218)(35 219)(36 220)(37 211)(38 212)(39 213)(40 214)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 240)(52 231)(53 232)(54 233)(55 234)(56 235)(57 236)(58 237)(59 238)(60 239)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 91)(81 167)(82 168)(83 169)(84 170)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 181)(112 182)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 190)(121 209)(122 210)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 207)(130 208)(151 156)(152 157)(153 158)(154 159)(155 160)(191 224)(192 225)(193 226)(194 227)(195 228)(196 229)(197 230)(198 221)(199 222)(200 223)
G:=sub<Sym(240)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240), (1,223,158,195)(2,224,159,196)(3,225,160,197)(4,226,151,198)(5,227,152,199)(6,228,153,200)(7,229,154,191)(8,230,155,192)(9,221,156,193)(10,222,157,194)(11,66,60,86)(12,67,51,87)(13,68,52,88)(14,69,53,89)(15,70,54,90)(16,61,55,81)(17,62,56,82)(18,63,57,83)(19,64,58,84)(20,65,59,85)(21,120,185,135)(22,111,186,136)(23,112,187,137)(24,113,188,138)(25,114,189,139)(26,115,190,140)(27,116,181,131)(28,117,182,132)(29,118,183,133)(30,119,184,134)(31,143,206,78)(32,144,207,79)(33,145,208,80)(34,146,209,71)(35,147,210,72)(36,148,201,73)(37,149,202,74)(38,150,203,75)(39,141,204,76)(40,142,205,77)(41,127,102,214)(42,128,103,215)(43,129,104,216)(44,130,105,217)(45,121,106,218)(46,122,107,219)(47,123,108,220)(48,124,109,211)(49,125,110,212)(50,126,101,213)(91,171,166,233)(92,172,167,234)(93,173,168,235)(94,174,169,236)(95,175,170,237)(96,176,161,238)(97,177,162,239)(98,178,163,240)(99,179,164,231)(100,180,165,232), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,211)(18,212)(19,213)(20,214)(31,162)(32,163)(33,164)(34,165)(35,166)(36,167)(37,168)(38,169)(39,170)(40,161)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,81)(48,82)(49,83)(50,84)(51,129)(52,130)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,108)(62,109)(63,110)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,232)(72,233)(73,234)(74,235)(75,236)(76,237)(77,238)(78,239)(79,240)(80,231)(91,210)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,209)(111,192)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,191)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,221)(138,222)(139,223)(140,224)(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,171)(148,172)(149,173)(150,174)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,181), (1,177,16)(2,178,17)(3,179,18)(4,180,19)(5,171,20)(6,172,11)(7,173,12)(8,174,13)(9,175,14)(10,176,15)(21,149,216)(22,150,217)(23,141,218)(24,142,219)(25,143,220)(26,144,211)(27,145,212)(28,146,213)(29,147,214)(30,148,215)(31,108,139)(32,109,140)(33,110,131)(34,101,132)(35,102,133)(36,103,134)(37,104,135)(38,105,136)(39,106,137)(40,107,138)(41,118,210)(42,119,201)(43,120,202)(44,111,203)(45,112,204)(46,113,205)(47,114,206)(48,115,207)(49,116,208)(50,117,209)(51,154,235)(52,155,236)(53,156,237)(54,157,238)(55,158,239)(56,159,240)(57,160,231)(58,151,232)(59,152,233)(60,153,234)(61,223,162)(62,224,163)(63,225,164)(64,226,165)(65,227,166)(66,228,167)(67,229,168)(68,230,169)(69,221,170)(70,222,161)(71,126,182)(72,127,183)(73,128,184)(74,129,185)(75,130,186)(76,121,187)(77,122,188)(78,123,189)(79,124,190)(80,125,181)(81,195,97)(82,196,98)(83,197,99)(84,198,100)(85,199,91)(86,200,92)(87,191,93)(88,192,94)(89,193,95)(90,194,96), (1,6)(2,7)(3,8)(4,9)(5,10)(11,177)(12,178)(13,179)(14,180)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,140)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,215)(32,216)(33,217)(34,218)(35,219)(36,220)(37,211)(38,212)(39,213)(40,214)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,240)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,91)(81,167)(82,168)(83,169)(84,170)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,209)(122,210)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(151,156)(152,157)(153,158)(154,159)(155,160)(191,224)(192,225)(193,226)(194,227)(195,228)(196,229)(197,230)(198,221)(199,222)(200,223)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240), (1,223,158,195)(2,224,159,196)(3,225,160,197)(4,226,151,198)(5,227,152,199)(6,228,153,200)(7,229,154,191)(8,230,155,192)(9,221,156,193)(10,222,157,194)(11,66,60,86)(12,67,51,87)(13,68,52,88)(14,69,53,89)(15,70,54,90)(16,61,55,81)(17,62,56,82)(18,63,57,83)(19,64,58,84)(20,65,59,85)(21,120,185,135)(22,111,186,136)(23,112,187,137)(24,113,188,138)(25,114,189,139)(26,115,190,140)(27,116,181,131)(28,117,182,132)(29,118,183,133)(30,119,184,134)(31,143,206,78)(32,144,207,79)(33,145,208,80)(34,146,209,71)(35,147,210,72)(36,148,201,73)(37,149,202,74)(38,150,203,75)(39,141,204,76)(40,142,205,77)(41,127,102,214)(42,128,103,215)(43,129,104,216)(44,130,105,217)(45,121,106,218)(46,122,107,219)(47,123,108,220)(48,124,109,211)(49,125,110,212)(50,126,101,213)(91,171,166,233)(92,172,167,234)(93,173,168,235)(94,174,169,236)(95,175,170,237)(96,176,161,238)(97,177,162,239)(98,178,163,240)(99,179,164,231)(100,180,165,232), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,211)(18,212)(19,213)(20,214)(31,162)(32,163)(33,164)(34,165)(35,166)(36,167)(37,168)(38,169)(39,170)(40,161)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,81)(48,82)(49,83)(50,84)(51,129)(52,130)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,108)(62,109)(63,110)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,232)(72,233)(73,234)(74,235)(75,236)(76,237)(77,238)(78,239)(79,240)(80,231)(91,210)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,209)(111,192)(112,193)(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,191)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,221)(138,222)(139,223)(140,224)(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,171)(148,172)(149,173)(150,174)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,181), (1,177,16)(2,178,17)(3,179,18)(4,180,19)(5,171,20)(6,172,11)(7,173,12)(8,174,13)(9,175,14)(10,176,15)(21,149,216)(22,150,217)(23,141,218)(24,142,219)(25,143,220)(26,144,211)(27,145,212)(28,146,213)(29,147,214)(30,148,215)(31,108,139)(32,109,140)(33,110,131)(34,101,132)(35,102,133)(36,103,134)(37,104,135)(38,105,136)(39,106,137)(40,107,138)(41,118,210)(42,119,201)(43,120,202)(44,111,203)(45,112,204)(46,113,205)(47,114,206)(48,115,207)(49,116,208)(50,117,209)(51,154,235)(52,155,236)(53,156,237)(54,157,238)(55,158,239)(56,159,240)(57,160,231)(58,151,232)(59,152,233)(60,153,234)(61,223,162)(62,224,163)(63,225,164)(64,226,165)(65,227,166)(66,228,167)(67,229,168)(68,230,169)(69,221,170)(70,222,161)(71,126,182)(72,127,183)(73,128,184)(74,129,185)(75,130,186)(76,121,187)(77,122,188)(78,123,189)(79,124,190)(80,125,181)(81,195,97)(82,196,98)(83,197,99)(84,198,100)(85,199,91)(86,200,92)(87,191,93)(88,192,94)(89,193,95)(90,194,96), (1,6)(2,7)(3,8)(4,9)(5,10)(11,177)(12,178)(13,179)(14,180)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,140)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,215)(32,216)(33,217)(34,218)(35,219)(36,220)(37,211)(38,212)(39,213)(40,214)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,240)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,91)(81,167)(82,168)(83,169)(84,170)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,209)(122,210)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(151,156)(152,157)(153,158)(154,159)(155,160)(191,224)(192,225)(193,226)(194,227)(195,228)(196,229)(197,230)(198,221)(199,222)(200,223) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190),(191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230),(231,232,233,234,235,236,237,238,239,240)], [(1,223,158,195),(2,224,159,196),(3,225,160,197),(4,226,151,198),(5,227,152,199),(6,228,153,200),(7,229,154,191),(8,230,155,192),(9,221,156,193),(10,222,157,194),(11,66,60,86),(12,67,51,87),(13,68,52,88),(14,69,53,89),(15,70,54,90),(16,61,55,81),(17,62,56,82),(18,63,57,83),(19,64,58,84),(20,65,59,85),(21,120,185,135),(22,111,186,136),(23,112,187,137),(24,113,188,138),(25,114,189,139),(26,115,190,140),(27,116,181,131),(28,117,182,132),(29,118,183,133),(30,119,184,134),(31,143,206,78),(32,144,207,79),(33,145,208,80),(34,146,209,71),(35,147,210,72),(36,148,201,73),(37,149,202,74),(38,150,203,75),(39,141,204,76),(40,142,205,77),(41,127,102,214),(42,128,103,215),(43,129,104,216),(44,130,105,217),(45,121,106,218),(46,122,107,219),(47,123,108,220),(48,124,109,211),(49,125,110,212),(50,126,101,213),(91,171,166,233),(92,172,167,234),(93,173,168,235),(94,174,169,236),(95,175,170,237),(96,176,161,238),(97,177,162,239),(98,178,163,240),(99,179,164,231),(100,180,165,232)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,21),(8,22),(9,23),(10,24),(11,215),(12,216),(13,217),(14,218),(15,219),(16,220),(17,211),(18,212),(19,213),(20,214),(31,162),(32,163),(33,164),(34,165),(35,166),(36,167),(37,168),(38,169),(39,170),(40,161),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,81),(48,82),(49,83),(50,84),(51,129),(52,130),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,108),(62,109),(63,110),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,232),(72,233),(73,234),(74,235),(75,236),(76,237),(77,238),(78,239),(79,240),(80,231),(91,210),(92,201),(93,202),(94,203),(95,204),(96,205),(97,206),(98,207),(99,208),(100,209),(111,192),(112,193),(113,194),(114,195),(115,196),(116,197),(117,198),(118,199),(119,200),(120,191),(131,225),(132,226),(133,227),(134,228),(135,229),(136,230),(137,221),(138,222),(139,223),(140,224),(141,175),(142,176),(143,177),(144,178),(145,179),(146,180),(147,171),(148,172),(149,173),(150,174),(151,182),(152,183),(153,184),(154,185),(155,186),(156,187),(157,188),(158,189),(159,190),(160,181)], [(1,177,16),(2,178,17),(3,179,18),(4,180,19),(5,171,20),(6,172,11),(7,173,12),(8,174,13),(9,175,14),(10,176,15),(21,149,216),(22,150,217),(23,141,218),(24,142,219),(25,143,220),(26,144,211),(27,145,212),(28,146,213),(29,147,214),(30,148,215),(31,108,139),(32,109,140),(33,110,131),(34,101,132),(35,102,133),(36,103,134),(37,104,135),(38,105,136),(39,106,137),(40,107,138),(41,118,210),(42,119,201),(43,120,202),(44,111,203),(45,112,204),(46,113,205),(47,114,206),(48,115,207),(49,116,208),(50,117,209),(51,154,235),(52,155,236),(53,156,237),(54,157,238),(55,158,239),(56,159,240),(57,160,231),(58,151,232),(59,152,233),(60,153,234),(61,223,162),(62,224,163),(63,225,164),(64,226,165),(65,227,166),(66,228,167),(67,229,168),(68,230,169),(69,221,170),(70,222,161),(71,126,182),(72,127,183),(73,128,184),(74,129,185),(75,130,186),(76,121,187),(77,122,188),(78,123,189),(79,124,190),(80,125,181),(81,195,97),(82,196,98),(83,197,99),(84,198,100),(85,199,91),(86,200,92),(87,191,93),(88,192,94),(89,193,95),(90,194,96)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,177),(12,178),(13,179),(14,180),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,140),(22,131),(23,132),(24,133),(25,134),(26,135),(27,136),(28,137),(29,138),(30,139),(31,215),(32,216),(33,217),(34,218),(35,219),(36,220),(37,211),(38,212),(39,213),(40,214),(41,77),(42,78),(43,79),(44,80),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,240),(52,231),(53,232),(54,233),(55,234),(56,235),(57,236),(58,237),(59,238),(60,239),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,91),(81,167),(82,168),(83,169),(84,170),(85,161),(86,162),(87,163),(88,164),(89,165),(90,166),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,181),(112,182),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,190),(121,209),(122,210),(123,201),(124,202),(125,203),(126,204),(127,205),(128,206),(129,207),(130,208),(151,156),(152,157),(153,158),(154,159),(155,160),(191,224),(192,225),(193,226),(194,227),(195,228),(196,229),(197,230),(198,221),(199,222),(200,223)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30AB | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×D8 | C5×C3⋊D4 | C5×C3⋊D4 | D4⋊S3 | C5×D4⋊S3 |
| kernel | C10×D4⋊S3 | C10×C3⋊C8 | C5×D4⋊S3 | C10×D12 | D4×C30 | C2×D4⋊S3 | C2×C3⋊C8 | D4⋊S3 | C2×D12 | C6×D4 | D4×C10 | C60 | C2×C30 | C2×C20 | C5×D4 | C30 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C6 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 16 | 8 | 8 | 2 | 8 |
Matrix representation of C10×D4⋊S3 ►in GL4(𝔽241) generated by
| 150 | 0 | 0 | 0 |
| 0 | 150 | 0 | 0 |
| 0 | 0 | 143 | 0 |
| 0 | 0 | 0 | 143 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 2 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 230 |
| 0 | 0 | 22 | 219 |
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,143,0,0,0,0,143],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,240,240],[1,0,0,0,0,1,0,0,0,0,22,22,0,0,230,219],[240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,1,1,0,0,0,0,240,0,0,0,1,1] >;
C10×D4⋊S3 in GAP, Magma, Sage, TeX
C_{10}\times D_4\rtimes S_3 % in TeX
G:=Group("C10xD4:S3"); // GroupNames label
G:=SmallGroup(480,810);
// by ID
G=gap.SmallGroup(480,810);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations