direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×Q8⋊3S3, C30.92C24, C60.239C23, Q8⋊6(S3×C10), (C5×Q8)⋊27D6, (C6×Q8)⋊6C10, D12⋊9(C2×C10), (Q8×C10)⋊17S3, (Q8×C30)⋊20C2, (C2×D12)⋊12C10, (C10×D12)⋊28C2, C30⋊19(C4○D4), (C2×C20).372D6, C6.9(C23×C10), (S3×C20)⋊24C22, (C5×D12)⋊39C22, C10.77(S3×C23), (Q8×C15)⋊33C22, D6.4(C22×C10), (S3×C10).39C23, (C2×C30).447C23, C12.23(C22×C10), (C2×C60).375C22, C20.212(C22×S3), (C5×Dic3).46C23, Dic3.10(C22×C10), (C10×Dic3).241C22, (S3×C2×C4)⋊5C10, C6⋊3(C5×C4○D4), C3⋊3(C10×C4○D4), (S3×C2×C20)⋊15C2, C15⋊28(C2×C4○D4), C4.23(S3×C2×C10), (C2×Q8)⋊8(C5×S3), (C4×S3)⋊5(C2×C10), (C3×Q8)⋊6(C2×C10), (C2×C4).62(S3×C10), C2.10(S3×C22×C10), C22.32(S3×C2×C10), (C2×C12).49(C2×C10), (S3×C2×C10).122C22, (C2×C6).67(C22×C10), (C22×S3).31(C2×C10), (C2×C10).379(C22×S3), (C2×Dic3).51(C2×C10), SmallGroup(480,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 676 in 328 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×2], C22, C22 [×12], C5, S3 [×6], C6, C6 [×2], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], Dic3 [×2], C12 [×6], D6 [×6], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×6], C20 [×2], C2×C10, C2×C10 [×12], C4×S3 [×12], D12 [×12], C2×Dic3, C2×C12 [×3], C3×Q8 [×4], C22×S3 [×3], C5×S3 [×6], C30, C30 [×2], C2×C4○D4, C2×C20 [×3], C2×C20 [×13], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], S3×C2×C4 [×3], C2×D12 [×3], Q8⋊3S3 [×8], C6×Q8, C5×Dic3 [×2], C60 [×6], S3×C10 [×6], S3×C10 [×6], C2×C30, C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×Q8⋊3S3, S3×C20 [×12], C5×D12 [×12], C10×Dic3, C2×C60 [×3], Q8×C15 [×4], S3×C2×C10 [×3], C10×C4○D4, S3×C2×C20 [×3], C10×D12 [×3], C5×Q8⋊3S3 [×8], Q8×C30, C10×Q8⋊3S3
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], Q8⋊3S3 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×Q8⋊3S3, S3×C2×C10 [×7], C10×C4○D4, C5×Q8⋊3S3 [×2], S3×C22×C10, C10×Q8⋊3S3
Generators and relations
G = < a,b,c,d,e | a10=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, 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)
(1 225 207 173)(2 226 208 174)(3 227 209 175)(4 228 210 176)(5 229 201 177)(6 230 202 178)(7 221 203 179)(8 222 204 180)(9 223 205 171)(10 224 206 172)(11 135 88 80)(12 136 89 71)(13 137 90 72)(14 138 81 73)(15 139 82 74)(16 140 83 75)(17 131 84 76)(18 132 85 77)(19 133 86 78)(20 134 87 79)(21 186 66 141)(22 187 67 142)(23 188 68 143)(24 189 69 144)(25 190 70 145)(26 181 61 146)(27 182 62 147)(28 183 63 148)(29 184 64 149)(30 185 65 150)(31 169 97 217)(32 170 98 218)(33 161 99 219)(34 162 100 220)(35 163 91 211)(36 164 92 212)(37 165 93 213)(38 166 94 214)(39 167 95 215)(40 168 96 216)(41 59 105 113)(42 60 106 114)(43 51 107 115)(44 52 108 116)(45 53 109 117)(46 54 110 118)(47 55 101 119)(48 56 102 120)(49 57 103 111)(50 58 104 112)(121 198 158 234)(122 199 159 235)(123 200 160 236)(124 191 151 237)(125 192 152 238)(126 193 153 239)(127 194 154 240)(128 195 155 231)(129 196 156 232)(130 197 157 233)
(1 237 207 191)(2 238 208 192)(3 239 209 193)(4 240 210 194)(5 231 201 195)(6 232 202 196)(7 233 203 197)(8 234 204 198)(9 235 205 199)(10 236 206 200)(11 112 88 58)(12 113 89 59)(13 114 90 60)(14 115 81 51)(15 116 82 52)(16 117 83 53)(17 118 84 54)(18 119 85 55)(19 120 86 56)(20 111 87 57)(21 92 66 36)(22 93 67 37)(23 94 68 38)(24 95 69 39)(25 96 70 40)(26 97 61 31)(27 98 62 32)(28 99 63 33)(29 100 64 34)(30 91 65 35)(41 136 105 71)(42 137 106 72)(43 138 107 73)(44 139 108 74)(45 140 109 75)(46 131 110 76)(47 132 101 77)(48 133 102 78)(49 134 103 79)(50 135 104 80)(121 222 158 180)(122 223 159 171)(123 224 160 172)(124 225 151 173)(125 226 152 174)(126 227 153 175)(127 228 154 176)(128 229 155 177)(129 230 156 178)(130 221 157 179)(141 212 186 164)(142 213 187 165)(143 214 188 166)(144 215 189 167)(145 216 190 168)(146 217 181 169)(147 218 182 170)(148 219 183 161)(149 220 184 162)(150 211 185 163)
(1 59 217)(2 60 218)(3 51 219)(4 52 220)(5 53 211)(6 54 212)(7 55 213)(8 56 214)(9 57 215)(10 58 216)(11 190 236)(12 181 237)(13 182 238)(14 183 239)(15 184 240)(16 185 231)(17 186 232)(18 187 233)(19 188 234)(20 189 235)(21 156 76)(22 157 77)(23 158 78)(24 159 79)(25 160 80)(26 151 71)(27 152 72)(28 153 73)(29 154 74)(30 155 75)(31 225 105)(32 226 106)(33 227 107)(34 228 108)(35 229 109)(36 230 110)(37 221 101)(38 222 102)(39 223 103)(40 224 104)(41 97 173)(42 98 174)(43 99 175)(44 100 176)(45 91 177)(46 92 178)(47 93 179)(48 94 180)(49 95 171)(50 96 172)(61 124 136)(62 125 137)(63 126 138)(64 127 139)(65 128 140)(66 129 131)(67 130 132)(68 121 133)(69 122 134)(70 123 135)(81 148 193)(82 149 194)(83 150 195)(84 141 196)(85 142 197)(86 143 198)(87 144 199)(88 145 200)(89 146 191)(90 147 192)(111 167 205)(112 168 206)(113 169 207)(114 170 208)(115 161 209)(116 162 210)(117 163 201)(118 164 202)(119 165 203)(120 166 204)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 159)(10 160)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 51)(29 52)(30 53)(31 89)(32 90)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 111)(70 112)(71 217)(72 218)(73 219)(74 220)(75 211)(76 212)(77 213)(78 214)(79 215)(80 216)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 141)(121 204)(122 205)(123 206)(124 207)(125 208)(126 209)(127 210)(128 201)(129 202)(130 203)(131 164)(132 165)(133 166)(134 167)(135 168)(136 169)(137 170)(138 161)(139 162)(140 163)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 231)(178 232)(179 233)(180 234)(191 225)(192 226)(193 227)(194 228)(195 229)(196 230)(197 221)(198 222)(199 223)(200 224)
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), (1,225,207,173)(2,226,208,174)(3,227,209,175)(4,228,210,176)(5,229,201,177)(6,230,202,178)(7,221,203,179)(8,222,204,180)(9,223,205,171)(10,224,206,172)(11,135,88,80)(12,136,89,71)(13,137,90,72)(14,138,81,73)(15,139,82,74)(16,140,83,75)(17,131,84,76)(18,132,85,77)(19,133,86,78)(20,134,87,79)(21,186,66,141)(22,187,67,142)(23,188,68,143)(24,189,69,144)(25,190,70,145)(26,181,61,146)(27,182,62,147)(28,183,63,148)(29,184,64,149)(30,185,65,150)(31,169,97,217)(32,170,98,218)(33,161,99,219)(34,162,100,220)(35,163,91,211)(36,164,92,212)(37,165,93,213)(38,166,94,214)(39,167,95,215)(40,168,96,216)(41,59,105,113)(42,60,106,114)(43,51,107,115)(44,52,108,116)(45,53,109,117)(46,54,110,118)(47,55,101,119)(48,56,102,120)(49,57,103,111)(50,58,104,112)(121,198,158,234)(122,199,159,235)(123,200,160,236)(124,191,151,237)(125,192,152,238)(126,193,153,239)(127,194,154,240)(128,195,155,231)(129,196,156,232)(130,197,157,233), (1,237,207,191)(2,238,208,192)(3,239,209,193)(4,240,210,194)(5,231,201,195)(6,232,202,196)(7,233,203,197)(8,234,204,198)(9,235,205,199)(10,236,206,200)(11,112,88,58)(12,113,89,59)(13,114,90,60)(14,115,81,51)(15,116,82,52)(16,117,83,53)(17,118,84,54)(18,119,85,55)(19,120,86,56)(20,111,87,57)(21,92,66,36)(22,93,67,37)(23,94,68,38)(24,95,69,39)(25,96,70,40)(26,97,61,31)(27,98,62,32)(28,99,63,33)(29,100,64,34)(30,91,65,35)(41,136,105,71)(42,137,106,72)(43,138,107,73)(44,139,108,74)(45,140,109,75)(46,131,110,76)(47,132,101,77)(48,133,102,78)(49,134,103,79)(50,135,104,80)(121,222,158,180)(122,223,159,171)(123,224,160,172)(124,225,151,173)(125,226,152,174)(126,227,153,175)(127,228,154,176)(128,229,155,177)(129,230,156,178)(130,221,157,179)(141,212,186,164)(142,213,187,165)(143,214,188,166)(144,215,189,167)(145,216,190,168)(146,217,181,169)(147,218,182,170)(148,219,183,161)(149,220,184,162)(150,211,185,163), (1,59,217)(2,60,218)(3,51,219)(4,52,220)(5,53,211)(6,54,212)(7,55,213)(8,56,214)(9,57,215)(10,58,216)(11,190,236)(12,181,237)(13,182,238)(14,183,239)(15,184,240)(16,185,231)(17,186,232)(18,187,233)(19,188,234)(20,189,235)(21,156,76)(22,157,77)(23,158,78)(24,159,79)(25,160,80)(26,151,71)(27,152,72)(28,153,73)(29,154,74)(30,155,75)(31,225,105)(32,226,106)(33,227,107)(34,228,108)(35,229,109)(36,230,110)(37,221,101)(38,222,102)(39,223,103)(40,224,104)(41,97,173)(42,98,174)(43,99,175)(44,100,176)(45,91,177)(46,92,178)(47,93,179)(48,94,180)(49,95,171)(50,96,172)(61,124,136)(62,125,137)(63,126,138)(64,127,139)(65,128,140)(66,129,131)(67,130,132)(68,121,133)(69,122,134)(70,123,135)(81,148,193)(82,149,194)(83,150,195)(84,141,196)(85,142,197)(86,143,198)(87,144,199)(88,145,200)(89,146,191)(90,147,192)(111,167,205)(112,168,206)(113,169,207)(114,170,208)(115,161,209)(116,162,210)(117,163,201)(118,164,202)(119,165,203)(120,166,204), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,89)(32,90)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,111)(70,112)(71,217)(72,218)(73,219)(74,220)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,141)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,201)(129,202)(130,203)(131,164)(132,165)(133,166)(134,167)(135,168)(136,169)(137,170)(138,161)(139,162)(140,163)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,231)(178,232)(179,233)(180,234)(191,225)(192,226)(193,227)(194,228)(195,229)(196,230)(197,221)(198,222)(199,223)(200,224)>;
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), (1,225,207,173)(2,226,208,174)(3,227,209,175)(4,228,210,176)(5,229,201,177)(6,230,202,178)(7,221,203,179)(8,222,204,180)(9,223,205,171)(10,224,206,172)(11,135,88,80)(12,136,89,71)(13,137,90,72)(14,138,81,73)(15,139,82,74)(16,140,83,75)(17,131,84,76)(18,132,85,77)(19,133,86,78)(20,134,87,79)(21,186,66,141)(22,187,67,142)(23,188,68,143)(24,189,69,144)(25,190,70,145)(26,181,61,146)(27,182,62,147)(28,183,63,148)(29,184,64,149)(30,185,65,150)(31,169,97,217)(32,170,98,218)(33,161,99,219)(34,162,100,220)(35,163,91,211)(36,164,92,212)(37,165,93,213)(38,166,94,214)(39,167,95,215)(40,168,96,216)(41,59,105,113)(42,60,106,114)(43,51,107,115)(44,52,108,116)(45,53,109,117)(46,54,110,118)(47,55,101,119)(48,56,102,120)(49,57,103,111)(50,58,104,112)(121,198,158,234)(122,199,159,235)(123,200,160,236)(124,191,151,237)(125,192,152,238)(126,193,153,239)(127,194,154,240)(128,195,155,231)(129,196,156,232)(130,197,157,233), (1,237,207,191)(2,238,208,192)(3,239,209,193)(4,240,210,194)(5,231,201,195)(6,232,202,196)(7,233,203,197)(8,234,204,198)(9,235,205,199)(10,236,206,200)(11,112,88,58)(12,113,89,59)(13,114,90,60)(14,115,81,51)(15,116,82,52)(16,117,83,53)(17,118,84,54)(18,119,85,55)(19,120,86,56)(20,111,87,57)(21,92,66,36)(22,93,67,37)(23,94,68,38)(24,95,69,39)(25,96,70,40)(26,97,61,31)(27,98,62,32)(28,99,63,33)(29,100,64,34)(30,91,65,35)(41,136,105,71)(42,137,106,72)(43,138,107,73)(44,139,108,74)(45,140,109,75)(46,131,110,76)(47,132,101,77)(48,133,102,78)(49,134,103,79)(50,135,104,80)(121,222,158,180)(122,223,159,171)(123,224,160,172)(124,225,151,173)(125,226,152,174)(126,227,153,175)(127,228,154,176)(128,229,155,177)(129,230,156,178)(130,221,157,179)(141,212,186,164)(142,213,187,165)(143,214,188,166)(144,215,189,167)(145,216,190,168)(146,217,181,169)(147,218,182,170)(148,219,183,161)(149,220,184,162)(150,211,185,163), (1,59,217)(2,60,218)(3,51,219)(4,52,220)(5,53,211)(6,54,212)(7,55,213)(8,56,214)(9,57,215)(10,58,216)(11,190,236)(12,181,237)(13,182,238)(14,183,239)(15,184,240)(16,185,231)(17,186,232)(18,187,233)(19,188,234)(20,189,235)(21,156,76)(22,157,77)(23,158,78)(24,159,79)(25,160,80)(26,151,71)(27,152,72)(28,153,73)(29,154,74)(30,155,75)(31,225,105)(32,226,106)(33,227,107)(34,228,108)(35,229,109)(36,230,110)(37,221,101)(38,222,102)(39,223,103)(40,224,104)(41,97,173)(42,98,174)(43,99,175)(44,100,176)(45,91,177)(46,92,178)(47,93,179)(48,94,180)(49,95,171)(50,96,172)(61,124,136)(62,125,137)(63,126,138)(64,127,139)(65,128,140)(66,129,131)(67,130,132)(68,121,133)(69,122,134)(70,123,135)(81,148,193)(82,149,194)(83,150,195)(84,141,196)(85,142,197)(86,143,198)(87,144,199)(88,145,200)(89,146,191)(90,147,192)(111,167,205)(112,168,206)(113,169,207)(114,170,208)(115,161,209)(116,162,210)(117,163,201)(118,164,202)(119,165,203)(120,166,204), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,89)(32,90)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,111)(70,112)(71,217)(72,218)(73,219)(74,220)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,141)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,201)(129,202)(130,203)(131,164)(132,165)(133,166)(134,167)(135,168)(136,169)(137,170)(138,161)(139,162)(140,163)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,231)(178,232)(179,233)(180,234)(191,225)(192,226)(193,227)(194,228)(195,229)(196,230)(197,221)(198,222)(199,223)(200,224) );
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)], [(1,225,207,173),(2,226,208,174),(3,227,209,175),(4,228,210,176),(5,229,201,177),(6,230,202,178),(7,221,203,179),(8,222,204,180),(9,223,205,171),(10,224,206,172),(11,135,88,80),(12,136,89,71),(13,137,90,72),(14,138,81,73),(15,139,82,74),(16,140,83,75),(17,131,84,76),(18,132,85,77),(19,133,86,78),(20,134,87,79),(21,186,66,141),(22,187,67,142),(23,188,68,143),(24,189,69,144),(25,190,70,145),(26,181,61,146),(27,182,62,147),(28,183,63,148),(29,184,64,149),(30,185,65,150),(31,169,97,217),(32,170,98,218),(33,161,99,219),(34,162,100,220),(35,163,91,211),(36,164,92,212),(37,165,93,213),(38,166,94,214),(39,167,95,215),(40,168,96,216),(41,59,105,113),(42,60,106,114),(43,51,107,115),(44,52,108,116),(45,53,109,117),(46,54,110,118),(47,55,101,119),(48,56,102,120),(49,57,103,111),(50,58,104,112),(121,198,158,234),(122,199,159,235),(123,200,160,236),(124,191,151,237),(125,192,152,238),(126,193,153,239),(127,194,154,240),(128,195,155,231),(129,196,156,232),(130,197,157,233)], [(1,237,207,191),(2,238,208,192),(3,239,209,193),(4,240,210,194),(5,231,201,195),(6,232,202,196),(7,233,203,197),(8,234,204,198),(9,235,205,199),(10,236,206,200),(11,112,88,58),(12,113,89,59),(13,114,90,60),(14,115,81,51),(15,116,82,52),(16,117,83,53),(17,118,84,54),(18,119,85,55),(19,120,86,56),(20,111,87,57),(21,92,66,36),(22,93,67,37),(23,94,68,38),(24,95,69,39),(25,96,70,40),(26,97,61,31),(27,98,62,32),(28,99,63,33),(29,100,64,34),(30,91,65,35),(41,136,105,71),(42,137,106,72),(43,138,107,73),(44,139,108,74),(45,140,109,75),(46,131,110,76),(47,132,101,77),(48,133,102,78),(49,134,103,79),(50,135,104,80),(121,222,158,180),(122,223,159,171),(123,224,160,172),(124,225,151,173),(125,226,152,174),(126,227,153,175),(127,228,154,176),(128,229,155,177),(129,230,156,178),(130,221,157,179),(141,212,186,164),(142,213,187,165),(143,214,188,166),(144,215,189,167),(145,216,190,168),(146,217,181,169),(147,218,182,170),(148,219,183,161),(149,220,184,162),(150,211,185,163)], [(1,59,217),(2,60,218),(3,51,219),(4,52,220),(5,53,211),(6,54,212),(7,55,213),(8,56,214),(9,57,215),(10,58,216),(11,190,236),(12,181,237),(13,182,238),(14,183,239),(15,184,240),(16,185,231),(17,186,232),(18,187,233),(19,188,234),(20,189,235),(21,156,76),(22,157,77),(23,158,78),(24,159,79),(25,160,80),(26,151,71),(27,152,72),(28,153,73),(29,154,74),(30,155,75),(31,225,105),(32,226,106),(33,227,107),(34,228,108),(35,229,109),(36,230,110),(37,221,101),(38,222,102),(39,223,103),(40,224,104),(41,97,173),(42,98,174),(43,99,175),(44,100,176),(45,91,177),(46,92,178),(47,93,179),(48,94,180),(49,95,171),(50,96,172),(61,124,136),(62,125,137),(63,126,138),(64,127,139),(65,128,140),(66,129,131),(67,130,132),(68,121,133),(69,122,134),(70,123,135),(81,148,193),(82,149,194),(83,150,195),(84,141,196),(85,142,197),(86,143,198),(87,144,199),(88,145,200),(89,146,191),(90,147,192),(111,167,205),(112,168,206),(113,169,207),(114,170,208),(115,161,209),(116,162,210),(117,163,201),(118,164,202),(119,165,203),(120,166,204)], [(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,159),(10,160),(11,96),(12,97),(13,98),(14,99),(15,100),(16,91),(17,92),(18,93),(19,94),(20,95),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,51),(29,52),(30,53),(31,89),(32,90),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,111),(70,112),(71,217),(72,218),(73,219),(74,220),(75,211),(76,212),(77,213),(78,214),(79,215),(80,216),(101,142),(102,143),(103,144),(104,145),(105,146),(106,147),(107,148),(108,149),(109,150),(110,141),(121,204),(122,205),(123,206),(124,207),(125,208),(126,209),(127,210),(128,201),(129,202),(130,203),(131,164),(132,165),(133,166),(134,167),(135,168),(136,169),(137,170),(138,161),(139,162),(140,163),(171,235),(172,236),(173,237),(174,238),(175,239),(176,240),(177,231),(178,232),(179,233),(180,234),(191,225),(192,226),(193,227),(194,228),(195,229),(196,230),(197,221),(198,222),(199,223),(200,224)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 52 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 3 |
| 0 | 0 | 35 | 4 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 44 | 28 |
| 0 | 0 | 42 | 17 |
G:=sub<GL(4,GF(61))| [3,0,0,0,0,3,0,0,0,0,20,0,0,0,0,20],[60,0,0,0,0,60,0,0,0,0,50,52,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,57,35,0,0,3,4],[0,60,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,44,42,0,0,28,17] >;
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AJ | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C4○D4 | C5×S3 | S3×C10 | S3×C10 | C5×C4○D4 | Q8⋊3S3 | C5×Q8⋊3S3 |
| kernel | C10×Q8⋊3S3 | S3×C2×C20 | C10×D12 | C5×Q8⋊3S3 | Q8×C30 | C2×Q8⋊3S3 | S3×C2×C4 | C2×D12 | Q8⋊3S3 | C6×Q8 | Q8×C10 | C2×C20 | C5×Q8 | C30 | C2×Q8 | C2×C4 | Q8 | C6 | C10 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 12 | 12 | 32 | 4 | 1 | 3 | 4 | 4 | 4 | 12 | 16 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times Q_8\rtimes_3S_3 % in TeX
G:=Group("C10xQ8:3S3"); // GroupNames label
G:=SmallGroup(480,1158);
// by ID
G=gap.SmallGroup(480,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,2467,633,304,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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