direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D4⋊2S3, C30.89C24, C60.236C23, (C6×D4)⋊6C10, (C5×D4)⋊27D6, D4⋊5(S3×C10), (D4×C10)⋊17S3, (D4×C30)⋊20C2, C30⋊16(C4○D4), Dic6⋊7(C2×C10), (C2×C20).370D6, C6.6(C23×C10), (S3×C20)⋊23C22, (C10×Dic6)⋊28C2, (C2×Dic6)⋊12C10, (D4×C15)⋊37C22, C23.24(S3×C10), C10.74(S3×C23), D6.2(C22×C10), (C22×C10).95D6, (S3×C10).37C23, C20.209(C22×S3), C12.20(C22×C10), (C2×C60).373C22, (C2×C30).445C23, (C22×Dic3)⋊8C10, (C5×Dic6)⋊34C22, (C10×Dic3)⋊36C22, Dic3.3(C22×C10), (C5×Dic3).39C23, (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, (C2×C10).21(C22×S3), (C22×C6).23(C2×C10), (C22×S3).30(C2×C10), SmallGroup(480,1155)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 612 in 328 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C10, C10 [×2], C10 [×6], Dic3 [×6], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×4], C2×C10 [×8], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C5×S3 [×2], C30, C30 [×2], C30 [×4], C2×C4○D4, C2×C20, C2×C20 [×15], C5×D4 [×4], C5×D4 [×8], C5×Q8 [×4], C22×C10 [×2], C22×C10, C2×Dic6, S3×C2×C4, D4⋊2S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×6], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C22×C20 [×3], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×8], C2×D4⋊2S3, C5×Dic6 [×4], S3×C20 [×4], C10×Dic3, C10×Dic3 [×10], C5×C3⋊D4 [×8], C2×C60, D4×C15 [×4], S3×C2×C10, C22×C30 [×2], C10×C4○D4, C10×Dic6, S3×C2×C20, C5×D4⋊2S3 [×8], Dic3×C2×C10 [×2], C10×C3⋊D4 [×2], D4×C30, C10×D4⋊2S3
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C4○D4 [×2], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C2×C4○D4, C22×C10 [×15], D4⋊2S3 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×D4⋊2S3, S3×C2×C10 [×7], C10×C4○D4, C5×D4⋊2S3 [×2], S3×C22×C10, C10×D4⋊2S3
Generators and relations
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 >
►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)
(1 153 93 163)(2 154 94 164)(3 155 95 165)(4 156 96 166)(5 157 97 167)(6 158 98 168)(7 159 99 169)(8 160 100 170)(9 151 91 161)(10 152 92 162)(11 230 29 220)(12 221 30 211)(13 222 21 212)(14 223 22 213)(15 224 23 214)(16 225 24 215)(17 226 25 216)(18 227 26 217)(19 228 27 218)(20 229 28 219)(31 190 53 202)(32 181 54 203)(33 182 55 204)(34 183 56 205)(35 184 57 206)(36 185 58 207)(37 186 59 208)(38 187 60 209)(39 188 51 210)(40 189 52 201)(41 200 237 177)(42 191 238 178)(43 192 239 179)(44 193 240 180)(45 194 231 171)(46 195 232 172)(47 196 233 173)(48 197 234 174)(49 198 235 175)(50 199 236 176)(61 143 83 121)(62 144 84 122)(63 145 85 123)(64 146 86 124)(65 147 87 125)(66 148 88 126)(67 149 89 127)(68 150 90 128)(69 141 81 129)(70 142 82 130)(71 118 107 131)(72 119 108 132)(73 120 109 133)(74 111 110 134)(75 112 101 135)(76 113 102 136)(77 114 103 137)(78 115 104 138)(79 116 105 139)(80 117 106 140)
(1 168)(2 169)(3 170)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 215)(12 216)(13 217)(14 218)(15 219)(16 220)(17 211)(18 212)(19 213)(20 214)(21 227)(22 228)(23 229)(24 230)(25 221)(26 222)(27 223)(28 224)(29 225)(30 226)(31 207)(32 208)(33 209)(34 210)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 172)(42 173)(43 174)(44 175)(45 176)(46 177)(47 178)(48 179)(49 180)(50 171)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 181)(60 182)(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)(101 117)(102 118)(103 119)(104 120)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)(191 233)(192 234)(193 235)(194 236)(195 237)(196 238)(197 239)(198 240)(199 231)(200 232)
(1 101 85)(2 102 86)(3 103 87)(4 104 88)(5 105 89)(6 106 90)(7 107 81)(8 108 82)(9 109 83)(10 110 84)(11 53 239)(12 54 240)(13 55 231)(14 56 232)(15 57 233)(16 58 234)(17 59 235)(18 60 236)(19 51 237)(20 52 238)(21 33 45)(22 34 46)(23 35 47)(24 36 48)(25 37 49)(26 38 50)(27 39 41)(28 40 42)(29 31 43)(30 32 44)(61 91 73)(62 92 74)(63 93 75)(64 94 76)(65 95 77)(66 96 78)(67 97 79)(68 98 80)(69 99 71)(70 100 72)(111 144 162)(112 145 163)(113 146 164)(114 147 165)(115 148 166)(116 149 167)(117 150 168)(118 141 169)(119 142 170)(120 143 161)(121 151 133)(122 152 134)(123 153 135)(124 154 136)(125 155 137)(126 156 138)(127 157 139)(128 158 140)(129 159 131)(130 160 132)(171 222 204)(172 223 205)(173 224 206)(174 225 207)(175 226 208)(176 227 209)(177 228 210)(178 229 201)(179 230 202)(180 221 203)(181 193 211)(182 194 212)(183 195 213)(184 196 214)(185 197 215)(186 198 216)(187 199 217)(188 200 218)(189 191 219)(190 192 220)
(1 223)(2 224)(3 225)(4 226)(5 227)(6 228)(7 229)(8 230)(9 221)(10 222)(11 170)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(31 130)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 140)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 150)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(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)(101 172)(102 173)(103 174)(104 175)(105 176)(106 177)(107 178)(108 179)(109 180)(110 171)(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), (1,153,93,163)(2,154,94,164)(3,155,95,165)(4,156,96,166)(5,157,97,167)(6,158,98,168)(7,159,99,169)(8,160,100,170)(9,151,91,161)(10,152,92,162)(11,230,29,220)(12,221,30,211)(13,222,21,212)(14,223,22,213)(15,224,23,214)(16,225,24,215)(17,226,25,216)(18,227,26,217)(19,228,27,218)(20,229,28,219)(31,190,53,202)(32,181,54,203)(33,182,55,204)(34,183,56,205)(35,184,57,206)(36,185,58,207)(37,186,59,208)(38,187,60,209)(39,188,51,210)(40,189,52,201)(41,200,237,177)(42,191,238,178)(43,192,239,179)(44,193,240,180)(45,194,231,171)(46,195,232,172)(47,196,233,173)(48,197,234,174)(49,198,235,175)(50,199,236,176)(61,143,83,121)(62,144,84,122)(63,145,85,123)(64,146,86,124)(65,147,87,125)(66,148,88,126)(67,149,89,127)(68,150,90,128)(69,141,81,129)(70,142,82,130)(71,118,107,131)(72,119,108,132)(73,120,109,133)(74,111,110,134)(75,112,101,135)(76,113,102,136)(77,114,103,137)(78,115,104,138)(79,116,105,139)(80,117,106,140), (1,168)(2,169)(3,170)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,211)(18,212)(19,213)(20,214)(21,227)(22,228)(23,229)(24,230)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,207)(32,208)(33,209)(34,210)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,172)(42,173)(43,174)(44,175)(45,176)(46,177)(47,178)(48,179)(49,180)(50,171)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,181)(60,182)(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)(101,117)(102,118)(103,119)(104,120)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,231)(200,232), (1,101,85)(2,102,86)(3,103,87)(4,104,88)(5,105,89)(6,106,90)(7,107,81)(8,108,82)(9,109,83)(10,110,84)(11,53,239)(12,54,240)(13,55,231)(14,56,232)(15,57,233)(16,58,234)(17,59,235)(18,60,236)(19,51,237)(20,52,238)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,41)(28,40,42)(29,31,43)(30,32,44)(61,91,73)(62,92,74)(63,93,75)(64,94,76)(65,95,77)(66,96,78)(67,97,79)(68,98,80)(69,99,71)(70,100,72)(111,144,162)(112,145,163)(113,146,164)(114,147,165)(115,148,166)(116,149,167)(117,150,168)(118,141,169)(119,142,170)(120,143,161)(121,151,133)(122,152,134)(123,153,135)(124,154,136)(125,155,137)(126,156,138)(127,157,139)(128,158,140)(129,159,131)(130,160,132)(171,222,204)(172,223,205)(173,224,206)(174,225,207)(175,226,208)(176,227,209)(177,228,210)(178,229,201)(179,230,202)(180,221,203)(181,193,211)(182,194,212)(183,195,213)(184,196,214)(185,197,215)(186,198,216)(187,199,217)(188,200,218)(189,191,219)(190,192,220), (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,221)(10,222)(11,170)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,169)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,150)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(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)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,171)(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), (1,153,93,163)(2,154,94,164)(3,155,95,165)(4,156,96,166)(5,157,97,167)(6,158,98,168)(7,159,99,169)(8,160,100,170)(9,151,91,161)(10,152,92,162)(11,230,29,220)(12,221,30,211)(13,222,21,212)(14,223,22,213)(15,224,23,214)(16,225,24,215)(17,226,25,216)(18,227,26,217)(19,228,27,218)(20,229,28,219)(31,190,53,202)(32,181,54,203)(33,182,55,204)(34,183,56,205)(35,184,57,206)(36,185,58,207)(37,186,59,208)(38,187,60,209)(39,188,51,210)(40,189,52,201)(41,200,237,177)(42,191,238,178)(43,192,239,179)(44,193,240,180)(45,194,231,171)(46,195,232,172)(47,196,233,173)(48,197,234,174)(49,198,235,175)(50,199,236,176)(61,143,83,121)(62,144,84,122)(63,145,85,123)(64,146,86,124)(65,147,87,125)(66,148,88,126)(67,149,89,127)(68,150,90,128)(69,141,81,129)(70,142,82,130)(71,118,107,131)(72,119,108,132)(73,120,109,133)(74,111,110,134)(75,112,101,135)(76,113,102,136)(77,114,103,137)(78,115,104,138)(79,116,105,139)(80,117,106,140), (1,168)(2,169)(3,170)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,211)(18,212)(19,213)(20,214)(21,227)(22,228)(23,229)(24,230)(25,221)(26,222)(27,223)(28,224)(29,225)(30,226)(31,207)(32,208)(33,209)(34,210)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,172)(42,173)(43,174)(44,175)(45,176)(46,177)(47,178)(48,179)(49,180)(50,171)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,181)(60,182)(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)(101,117)(102,118)(103,119)(104,120)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(191,233)(192,234)(193,235)(194,236)(195,237)(196,238)(197,239)(198,240)(199,231)(200,232), (1,101,85)(2,102,86)(3,103,87)(4,104,88)(5,105,89)(6,106,90)(7,107,81)(8,108,82)(9,109,83)(10,110,84)(11,53,239)(12,54,240)(13,55,231)(14,56,232)(15,57,233)(16,58,234)(17,59,235)(18,60,236)(19,51,237)(20,52,238)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,41)(28,40,42)(29,31,43)(30,32,44)(61,91,73)(62,92,74)(63,93,75)(64,94,76)(65,95,77)(66,96,78)(67,97,79)(68,98,80)(69,99,71)(70,100,72)(111,144,162)(112,145,163)(113,146,164)(114,147,165)(115,148,166)(116,149,167)(117,150,168)(118,141,169)(119,142,170)(120,143,161)(121,151,133)(122,152,134)(123,153,135)(124,154,136)(125,155,137)(126,156,138)(127,157,139)(128,158,140)(129,159,131)(130,160,132)(171,222,204)(172,223,205)(173,224,206)(174,225,207)(175,226,208)(176,227,209)(177,228,210)(178,229,201)(179,230,202)(180,221,203)(181,193,211)(182,194,212)(183,195,213)(184,196,214)(185,197,215)(186,198,216)(187,199,217)(188,200,218)(189,191,219)(190,192,220), (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,221)(10,222)(11,170)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,169)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,150)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(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)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,171)(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)], [(1,153,93,163),(2,154,94,164),(3,155,95,165),(4,156,96,166),(5,157,97,167),(6,158,98,168),(7,159,99,169),(8,160,100,170),(9,151,91,161),(10,152,92,162),(11,230,29,220),(12,221,30,211),(13,222,21,212),(14,223,22,213),(15,224,23,214),(16,225,24,215),(17,226,25,216),(18,227,26,217),(19,228,27,218),(20,229,28,219),(31,190,53,202),(32,181,54,203),(33,182,55,204),(34,183,56,205),(35,184,57,206),(36,185,58,207),(37,186,59,208),(38,187,60,209),(39,188,51,210),(40,189,52,201),(41,200,237,177),(42,191,238,178),(43,192,239,179),(44,193,240,180),(45,194,231,171),(46,195,232,172),(47,196,233,173),(48,197,234,174),(49,198,235,175),(50,199,236,176),(61,143,83,121),(62,144,84,122),(63,145,85,123),(64,146,86,124),(65,147,87,125),(66,148,88,126),(67,149,89,127),(68,150,90,128),(69,141,81,129),(70,142,82,130),(71,118,107,131),(72,119,108,132),(73,120,109,133),(74,111,110,134),(75,112,101,135),(76,113,102,136),(77,114,103,137),(78,115,104,138),(79,116,105,139),(80,117,106,140)], [(1,168),(2,169),(3,170),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,215),(12,216),(13,217),(14,218),(15,219),(16,220),(17,211),(18,212),(19,213),(20,214),(21,227),(22,228),(23,229),(24,230),(25,221),(26,222),(27,223),(28,224),(29,225),(30,226),(31,207),(32,208),(33,209),(34,210),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,172),(42,173),(43,174),(44,175),(45,176),(46,177),(47,178),(48,179),(49,180),(50,171),(51,183),(52,184),(53,185),(54,186),(55,187),(56,188),(57,189),(58,190),(59,181),(60,182),(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),(101,117),(102,118),(103,119),(104,120),(105,111),(106,112),(107,113),(108,114),(109,115),(110,116),(191,233),(192,234),(193,235),(194,236),(195,237),(196,238),(197,239),(198,240),(199,231),(200,232)], [(1,101,85),(2,102,86),(3,103,87),(4,104,88),(5,105,89),(6,106,90),(7,107,81),(8,108,82),(9,109,83),(10,110,84),(11,53,239),(12,54,240),(13,55,231),(14,56,232),(15,57,233),(16,58,234),(17,59,235),(18,60,236),(19,51,237),(20,52,238),(21,33,45),(22,34,46),(23,35,47),(24,36,48),(25,37,49),(26,38,50),(27,39,41),(28,40,42),(29,31,43),(30,32,44),(61,91,73),(62,92,74),(63,93,75),(64,94,76),(65,95,77),(66,96,78),(67,97,79),(68,98,80),(69,99,71),(70,100,72),(111,144,162),(112,145,163),(113,146,164),(114,147,165),(115,148,166),(116,149,167),(117,150,168),(118,141,169),(119,142,170),(120,143,161),(121,151,133),(122,152,134),(123,153,135),(124,154,136),(125,155,137),(126,156,138),(127,157,139),(128,158,140),(129,159,131),(130,160,132),(171,222,204),(172,223,205),(173,224,206),(174,225,207),(175,226,208),(176,227,209),(177,228,210),(178,229,201),(179,230,202),(180,221,203),(181,193,211),(182,194,212),(183,195,213),(184,196,214),(185,197,215),(186,198,216),(187,199,217),(188,200,218),(189,191,219),(190,192,220)], [(1,223),(2,224),(3,225),(4,226),(5,227),(6,228),(7,229),(8,230),(9,221),(10,222),(11,170),(12,161),(13,162),(14,163),(15,164),(16,165),(17,166),(18,167),(19,168),(20,169),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(31,130),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,140),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,150),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(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),(101,172),(102,173),(103,174),(104,175),(105,176),(106,177),(107,178),(108,179),(109,180),(110,171),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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