direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D4⋊2S3, C30.89C24, C60.236C23, (C5×D4)⋊27D6, (C6×D4)⋊6C10, D4⋊5(S3×C10), (D4×C30)⋊20C2, (D4×C10)⋊17S3, C30⋊16(C4○D4), Dic6⋊7(C2×C10), (C2×C20).370D6, C6.6(C23×C10), (S3×C20)⋊23C22, (C2×Dic6)⋊12C10, (C10×Dic6)⋊28C2, (D4×C15)⋊37C22, C23.24(S3×C10), C10.74(S3×C23), D6.2(C22×C10), (C22×C10).95D6, (S3×C10).37C23, (C2×C60).373C22, C12.20(C22×C10), C20.209(C22×S3), (C2×C30).445C23, (C22×Dic3)⋊8C10, (C5×Dic6)⋊34C22, (C10×Dic3)⋊36C22, (C5×Dic3).39C23, Dic3.3(C22×C10), (C22×C30).128C22, (S3×C2×C4)⋊4C10, C6⋊2(C5×C4○D4), C3⋊2(C10×C4○D4), (S3×C2×C20)⋊14C2, (C2×D4)⋊8(C5×S3), C15⋊25(C2×C4○D4), C4.20(S3×C2×C10), (C4×S3)⋊4(C2×C10), (C3×D4)⋊6(C2×C10), C3⋊D4⋊2(C2×C10), C22.1(S3×C2×C10), C2.7(S3×C22×C10), (C2×C3⋊D4)⋊10C10, (C10×C3⋊D4)⋊25C2, (C2×C4).60(S3×C10), (Dic3×C2×C10)⋊19C2, (C2×C12).47(C2×C10), (C2×Dic3)⋊9(C2×C10), (C5×C3⋊D4)⋊18C22, (C2×C6).1(C22×C10), (S3×C2×C10).121C22, (C22×C6).23(C2×C10), (C2×C10).21(C22×S3), (C22×S3).30(C2×C10), SmallGroup(480,1155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D4⋊2S3
G = < a,b,c,d,e | a10=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 612 in 328 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C30, C30, C30, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C22×C10, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C2×D4⋊2S3, C5×Dic6, S3×C20, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, S3×C2×C10, C22×C30, C10×C4○D4, C10×Dic6, S3×C2×C20, C5×D4⋊2S3, Dic3×C2×C10, C10×C3⋊D4, D4×C30, C10×D4⋊2S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C24, C2×C10, C22×S3, C5×S3, C2×C4○D4, C22×C10, D4⋊2S3, S3×C23, S3×C10, C5×C4○D4, C23×C10, C2×D4⋊2S3, S3×C2×C10, C10×C4○D4, C5×D4⋊2S3, S3×C22×C10, C10×D4⋊2S3
►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)
(1 165 105 116)(2 166 106 117)(3 167 107 118)(4 168 108 119)(5 169 109 120)(6 170 110 111)(7 161 101 112)(8 162 102 113)(9 163 103 114)(10 164 104 115)(11 218 31 210)(12 219 32 201)(13 220 33 202)(14 211 34 203)(15 212 35 204)(16 213 36 205)(17 214 37 206)(18 215 38 207)(19 216 39 208)(20 217 40 209)(21 224 235 175)(22 225 236 176)(23 226 237 177)(24 227 238 178)(25 228 239 179)(26 229 240 180)(27 230 231 171)(28 221 232 172)(29 222 233 173)(30 223 234 174)(41 200 55 186)(42 191 56 187)(43 192 57 188)(44 193 58 189)(45 194 59 190)(46 195 60 181)(47 196 51 182)(48 197 52 183)(49 198 53 184)(50 199 54 185)(61 135 75 121)(62 136 76 122)(63 137 77 123)(64 138 78 124)(65 139 79 125)(66 140 80 126)(67 131 71 127)(68 132 72 128)(69 133 73 129)(70 134 74 130)(81 159 99 141)(82 160 100 142)(83 151 91 143)(84 152 92 144)(85 153 93 145)(86 154 94 146)(87 155 95 147)(88 156 96 148)(89 157 97 149)(90 158 98 150)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 205)(12 206)(13 207)(14 208)(15 209)(16 210)(17 201)(18 202)(19 203)(20 204)(21 180)(22 171)(23 172)(24 173)(25 174)(26 175)(27 176)(28 177)(29 178)(30 179)(31 213)(32 214)(33 215)(34 216)(35 217)(36 218)(37 219)(38 220)(39 211)(40 212)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 197)(58 198)(59 199)(60 200)(61 126)(62 127)(63 128)(64 129)(65 130)(66 121)(67 122)(68 123)(69 124)(70 125)(71 136)(72 137)(73 138)(74 139)(75 140)(76 131)(77 132)(78 133)(79 134)(80 135)(81 146)(82 147)(83 148)(84 149)(85 150)(86 141)(87 142)(88 143)(89 144)(90 145)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(101 166)(102 167)(103 168)(104 169)(105 170)(106 161)(107 162)(108 163)(109 164)(110 165)(221 237)(222 238)(223 239)(224 240)(225 231)(226 232)(227 233)(228 234)(229 235)(230 236)
(1 61 97)(2 62 98)(3 63 99)(4 64 100)(5 65 91)(6 66 92)(7 67 93)(8 68 94)(9 69 95)(10 70 96)(11 51 237)(12 52 238)(13 53 239)(14 54 240)(15 55 231)(16 56 232)(17 57 233)(18 58 234)(19 59 235)(20 60 236)(21 39 45)(22 40 46)(23 31 47)(24 32 48)(25 33 49)(26 34 50)(27 35 41)(28 36 42)(29 37 43)(30 38 44)(71 85 101)(72 86 102)(73 87 103)(74 88 104)(75 89 105)(76 90 106)(77 81 107)(78 82 108)(79 83 109)(80 84 110)(111 126 152)(112 127 153)(113 128 154)(114 129 155)(115 130 156)(116 121 157)(117 122 158)(118 123 159)(119 124 160)(120 125 151)(131 145 161)(132 146 162)(133 147 163)(134 148 164)(135 149 165)(136 150 166)(137 141 167)(138 142 168)(139 143 169)(140 144 170)(171 212 186)(172 213 187)(173 214 188)(174 215 189)(175 216 190)(176 217 181)(177 218 182)(178 219 183)(179 220 184)(180 211 185)(191 221 205)(192 222 206)(193 223 207)(194 224 208)(195 225 209)(196 226 210)(197 227 201)(198 228 202)(199 229 203)(200 230 204)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 171)(7 172)(8 173)(9 174)(10 175)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 161)(29 162)(30 163)(31 150)(32 141)(33 142)(34 143)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(41 140)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 122)(52 123)(53 124)(54 125)(55 126)(56 127)(57 128)(58 129)(59 130)(60 121)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)
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), (1,165,105,116)(2,166,106,117)(3,167,107,118)(4,168,108,119)(5,169,109,120)(6,170,110,111)(7,161,101,112)(8,162,102,113)(9,163,103,114)(10,164,104,115)(11,218,31,210)(12,219,32,201)(13,220,33,202)(14,211,34,203)(15,212,35,204)(16,213,36,205)(17,214,37,206)(18,215,38,207)(19,216,39,208)(20,217,40,209)(21,224,235,175)(22,225,236,176)(23,226,237,177)(24,227,238,178)(25,228,239,179)(26,229,240,180)(27,230,231,171)(28,221,232,172)(29,222,233,173)(30,223,234,174)(41,200,55,186)(42,191,56,187)(43,192,57,188)(44,193,58,189)(45,194,59,190)(46,195,60,181)(47,196,51,182)(48,197,52,183)(49,198,53,184)(50,199,54,185)(61,135,75,121)(62,136,76,122)(63,137,77,123)(64,138,78,124)(65,139,79,125)(66,140,80,126)(67,131,71,127)(68,132,72,128)(69,133,73,129)(70,134,74,130)(81,159,99,141)(82,160,100,142)(83,151,91,143)(84,152,92,144)(85,153,93,145)(86,154,94,146)(87,155,95,147)(88,156,96,148)(89,157,97,149)(90,158,98,150), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,201)(18,202)(19,203)(20,204)(21,180)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,211)(40,212)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,197)(58,198)(59,199)(60,200)(61,126)(62,127)(63,128)(64,129)(65,130)(66,121)(67,122)(68,123)(69,124)(70,125)(71,136)(72,137)(73,138)(74,139)(75,140)(76,131)(77,132)(78,133)(79,134)(80,135)(81,146)(82,147)(83,148)(84,149)(85,150)(86,141)(87,142)(88,143)(89,144)(90,145)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(101,166)(102,167)(103,168)(104,169)(105,170)(106,161)(107,162)(108,163)(109,164)(110,165)(221,237)(222,238)(223,239)(224,240)(225,231)(226,232)(227,233)(228,234)(229,235)(230,236), (1,61,97)(2,62,98)(3,63,99)(4,64,100)(5,65,91)(6,66,92)(7,67,93)(8,68,94)(9,69,95)(10,70,96)(11,51,237)(12,52,238)(13,53,239)(14,54,240)(15,55,231)(16,56,232)(17,57,233)(18,58,234)(19,59,235)(20,60,236)(21,39,45)(22,40,46)(23,31,47)(24,32,48)(25,33,49)(26,34,50)(27,35,41)(28,36,42)(29,37,43)(30,38,44)(71,85,101)(72,86,102)(73,87,103)(74,88,104)(75,89,105)(76,90,106)(77,81,107)(78,82,108)(79,83,109)(80,84,110)(111,126,152)(112,127,153)(113,128,154)(114,129,155)(115,130,156)(116,121,157)(117,122,158)(118,123,159)(119,124,160)(120,125,151)(131,145,161)(132,146,162)(133,147,163)(134,148,164)(135,149,165)(136,150,166)(137,141,167)(138,142,168)(139,143,169)(140,144,170)(171,212,186)(172,213,187)(173,214,188)(174,215,189)(175,216,190)(176,217,181)(177,218,182)(178,219,183)(179,220,184)(180,211,185)(191,221,205)(192,222,206)(193,223,207)(194,224,208)(195,225,209)(196,226,210)(197,227,201)(198,228,202)(199,229,203)(200,230,204), (1,176)(2,177)(3,178)(4,179)(5,180)(6,171)(7,172)(8,173)(9,174)(10,175)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,161)(29,162)(30,163)(31,150)(32,141)(33,142)(34,143)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,121)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)>;
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), (1,165,105,116)(2,166,106,117)(3,167,107,118)(4,168,108,119)(5,169,109,120)(6,170,110,111)(7,161,101,112)(8,162,102,113)(9,163,103,114)(10,164,104,115)(11,218,31,210)(12,219,32,201)(13,220,33,202)(14,211,34,203)(15,212,35,204)(16,213,36,205)(17,214,37,206)(18,215,38,207)(19,216,39,208)(20,217,40,209)(21,224,235,175)(22,225,236,176)(23,226,237,177)(24,227,238,178)(25,228,239,179)(26,229,240,180)(27,230,231,171)(28,221,232,172)(29,222,233,173)(30,223,234,174)(41,200,55,186)(42,191,56,187)(43,192,57,188)(44,193,58,189)(45,194,59,190)(46,195,60,181)(47,196,51,182)(48,197,52,183)(49,198,53,184)(50,199,54,185)(61,135,75,121)(62,136,76,122)(63,137,77,123)(64,138,78,124)(65,139,79,125)(66,140,80,126)(67,131,71,127)(68,132,72,128)(69,133,73,129)(70,134,74,130)(81,159,99,141)(82,160,100,142)(83,151,91,143)(84,152,92,144)(85,153,93,145)(86,154,94,146)(87,155,95,147)(88,156,96,148)(89,157,97,149)(90,158,98,150), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,201)(18,202)(19,203)(20,204)(21,180)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,211)(40,212)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,197)(58,198)(59,199)(60,200)(61,126)(62,127)(63,128)(64,129)(65,130)(66,121)(67,122)(68,123)(69,124)(70,125)(71,136)(72,137)(73,138)(74,139)(75,140)(76,131)(77,132)(78,133)(79,134)(80,135)(81,146)(82,147)(83,148)(84,149)(85,150)(86,141)(87,142)(88,143)(89,144)(90,145)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(101,166)(102,167)(103,168)(104,169)(105,170)(106,161)(107,162)(108,163)(109,164)(110,165)(221,237)(222,238)(223,239)(224,240)(225,231)(226,232)(227,233)(228,234)(229,235)(230,236), (1,61,97)(2,62,98)(3,63,99)(4,64,100)(5,65,91)(6,66,92)(7,67,93)(8,68,94)(9,69,95)(10,70,96)(11,51,237)(12,52,238)(13,53,239)(14,54,240)(15,55,231)(16,56,232)(17,57,233)(18,58,234)(19,59,235)(20,60,236)(21,39,45)(22,40,46)(23,31,47)(24,32,48)(25,33,49)(26,34,50)(27,35,41)(28,36,42)(29,37,43)(30,38,44)(71,85,101)(72,86,102)(73,87,103)(74,88,104)(75,89,105)(76,90,106)(77,81,107)(78,82,108)(79,83,109)(80,84,110)(111,126,152)(112,127,153)(113,128,154)(114,129,155)(115,130,156)(116,121,157)(117,122,158)(118,123,159)(119,124,160)(120,125,151)(131,145,161)(132,146,162)(133,147,163)(134,148,164)(135,149,165)(136,150,166)(137,141,167)(138,142,168)(139,143,169)(140,144,170)(171,212,186)(172,213,187)(173,214,188)(174,215,189)(175,216,190)(176,217,181)(177,218,182)(178,219,183)(179,220,184)(180,211,185)(191,221,205)(192,222,206)(193,223,207)(194,224,208)(195,225,209)(196,226,210)(197,227,201)(198,228,202)(199,229,203)(200,230,204), (1,176)(2,177)(3,178)(4,179)(5,180)(6,171)(7,172)(8,173)(9,174)(10,175)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,161)(29,162)(30,163)(31,150)(32,141)(33,142)(34,143)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,121)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240) );
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)], [(1,165,105,116),(2,166,106,117),(3,167,107,118),(4,168,108,119),(5,169,109,120),(6,170,110,111),(7,161,101,112),(8,162,102,113),(9,163,103,114),(10,164,104,115),(11,218,31,210),(12,219,32,201),(13,220,33,202),(14,211,34,203),(15,212,35,204),(16,213,36,205),(17,214,37,206),(18,215,38,207),(19,216,39,208),(20,217,40,209),(21,224,235,175),(22,225,236,176),(23,226,237,177),(24,227,238,178),(25,228,239,179),(26,229,240,180),(27,230,231,171),(28,221,232,172),(29,222,233,173),(30,223,234,174),(41,200,55,186),(42,191,56,187),(43,192,57,188),(44,193,58,189),(45,194,59,190),(46,195,60,181),(47,196,51,182),(48,197,52,183),(49,198,53,184),(50,199,54,185),(61,135,75,121),(62,136,76,122),(63,137,77,123),(64,138,78,124),(65,139,79,125),(66,140,80,126),(67,131,71,127),(68,132,72,128),(69,133,73,129),(70,134,74,130),(81,159,99,141),(82,160,100,142),(83,151,91,143),(84,152,92,144),(85,153,93,145),(86,154,94,146),(87,155,95,147),(88,156,96,148),(89,157,97,149),(90,158,98,150)], [(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,205),(12,206),(13,207),(14,208),(15,209),(16,210),(17,201),(18,202),(19,203),(20,204),(21,180),(22,171),(23,172),(24,173),(25,174),(26,175),(27,176),(28,177),(29,178),(30,179),(31,213),(32,214),(33,215),(34,216),(35,217),(36,218),(37,219),(38,220),(39,211),(40,212),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,197),(58,198),(59,199),(60,200),(61,126),(62,127),(63,128),(64,129),(65,130),(66,121),(67,122),(68,123),(69,124),(70,125),(71,136),(72,137),(73,138),(74,139),(75,140),(76,131),(77,132),(78,133),(79,134),(80,135),(81,146),(82,147),(83,148),(84,149),(85,150),(86,141),(87,142),(88,143),(89,144),(90,145),(91,156),(92,157),(93,158),(94,159),(95,160),(96,151),(97,152),(98,153),(99,154),(100,155),(101,166),(102,167),(103,168),(104,169),(105,170),(106,161),(107,162),(108,163),(109,164),(110,165),(221,237),(222,238),(223,239),(224,240),(225,231),(226,232),(227,233),(228,234),(229,235),(230,236)], [(1,61,97),(2,62,98),(3,63,99),(4,64,100),(5,65,91),(6,66,92),(7,67,93),(8,68,94),(9,69,95),(10,70,96),(11,51,237),(12,52,238),(13,53,239),(14,54,240),(15,55,231),(16,56,232),(17,57,233),(18,58,234),(19,59,235),(20,60,236),(21,39,45),(22,40,46),(23,31,47),(24,32,48),(25,33,49),(26,34,50),(27,35,41),(28,36,42),(29,37,43),(30,38,44),(71,85,101),(72,86,102),(73,87,103),(74,88,104),(75,89,105),(76,90,106),(77,81,107),(78,82,108),(79,83,109),(80,84,110),(111,126,152),(112,127,153),(113,128,154),(114,129,155),(115,130,156),(116,121,157),(117,122,158),(118,123,159),(119,124,160),(120,125,151),(131,145,161),(132,146,162),(133,147,163),(134,148,164),(135,149,165),(136,150,166),(137,141,167),(138,142,168),(139,143,169),(140,144,170),(171,212,186),(172,213,187),(173,214,188),(174,215,189),(175,216,190),(176,217,181),(177,218,182),(178,219,183),(179,220,184),(180,211,185),(191,221,205),(192,222,206),(193,223,207),(194,224,208),(195,225,209),(196,226,210),(197,227,201),(198,228,202),(199,229,203),(200,230,204)], [(1,176),(2,177),(3,178),(4,179),(5,180),(6,171),(7,172),(8,173),(9,174),(10,175),(11,158),(12,159),(13,160),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,157),(21,164),(22,165),(23,166),(24,167),(25,168),(26,169),(27,170),(28,161),(29,162),(30,163),(31,150),(32,141),(33,142),(34,143),(35,144),(36,145),(37,146),(38,147),(39,148),(40,149),(41,140),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,122),(52,123),(53,124),(54,125),(55,126),(56,127),(57,128),(58,129),(59,130),(60,121),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10L | 10M | ··· | 10AB | 10AC | ··· | 10AJ | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | D6 | C4○D4 | C5×S3 | S3×C10 | S3×C10 | S3×C10 | C5×C4○D4 | D4⋊2S3 | C5×D4⋊2S3 |
| kernel | C10×D4⋊2S3 | C10×Dic6 | S3×C2×C20 | C5×D4⋊2S3 | Dic3×C2×C10 | C10×C3⋊D4 | D4×C30 | C2×D4⋊2S3 | C2×Dic6 | S3×C2×C4 | D4⋊2S3 | C22×Dic3 | C2×C3⋊D4 | C6×D4 | D4×C10 | C2×C20 | C5×D4 | C22×C10 | C30 | C2×D4 | C2×C4 | D4 | C23 | C6 | C10 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 4 | 4 | 4 | 32 | 8 | 8 | 4 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 16 | 8 | 16 | 2 | 8 |
Matrix representation of C10×D4⋊2S3 ►in GL5(𝔽61)
| 41 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 40 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 19 | 31 | 0 | 0 |
| 0 | 12 | 42 | 0 | 0 |
| 0 | 0 | 0 | 50 | 28 |
| 0 | 0 | 0 | 48 | 11 |
G:=sub<GL(5,GF(61))| [41,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,9,0,0,0,0,0,9],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,3,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3,60],[1,0,0,0,0,0,0,60,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,19,12,0,0,0,31,42,0,0,0,0,0,50,48,0,0,0,28,11] >;
C10×D4⋊2S3 in GAP, Magma, Sage, TeX
C_{10}\times D_4\rtimes_2S_3 % in TeX
G:=Group("C10xD4:2S3"); // GroupNames label
G:=SmallGroup(480,1155);
// by ID
G=gap.SmallGroup(480,1155);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,2467,633,15686]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations