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G = C10×Q8⋊2S3order 480 = 25·3·5

Direct product of C10 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×Q8⋊2S3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C10×Q8⋊2S3
 Lower central C3 — C6 — C12 — C10×Q8⋊2S3
 Upper central C1 — C2×C10 — C2×C20 — Q8×C10

Generators and relations for C10×Q82S3
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, ece=b-1c, ede=d-1 >

Subgroups: 356 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C10, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C5×S3, C30, C30, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×C3⋊C8, Q82S3, C2×D12, C6×Q8, C60, C60, S3×C10, C2×C30, C2×C40, C5×SD16, D4×C10, Q8×C10, C2×Q82S3, C5×C3⋊C8, C5×D12, C5×D12, C2×C60, C2×C60, Q8×C15, Q8×C15, S3×C2×C10, C10×SD16, C10×C3⋊C8, C5×Q82S3, C10×D12, Q8×C30, C10×Q82S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, SD16, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C2×SD16, C5×D4, C22×C10, Q82S3, C2×C3⋊D4, S3×C10, C5×SD16, D4×C10, C2×Q82S3, C5×C3⋊D4, S3×C2×C10, C10×SD16, C5×Q82S3, C10×C3⋊D4, C10×Q82S3

Smallest permutation representation of C10×Q82S3
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)
(1 197 38 185)(2 198 39 186)(3 199 40 187)(4 200 31 188)(5 191 32 189)(6 192 33 190)(7 193 34 181)(8 194 35 182)(9 195 36 183)(10 196 37 184)(11 159 206 116)(12 160 207 117)(13 151 208 118)(14 152 209 119)(15 153 210 120)(16 154 201 111)(17 155 202 112)(18 156 203 113)(19 157 204 114)(20 158 205 115)(21 230 78 123)(22 221 79 124)(23 222 80 125)(24 223 71 126)(25 224 72 127)(26 225 73 128)(27 226 74 129)(28 227 75 130)(29 228 76 121)(30 229 77 122)(41 141 98 55)(42 142 99 56)(43 143 100 57)(44 144 91 58)(45 145 92 59)(46 146 93 60)(47 147 94 51)(48 148 95 52)(49 149 96 53)(50 150 97 54)(61 136 239 101)(62 137 240 102)(63 138 231 103)(64 139 232 104)(65 140 233 105)(66 131 234 106)(67 132 235 107)(68 133 236 108)(69 134 237 109)(70 135 238 110)(81 162 217 177)(82 163 218 178)(83 164 219 179)(84 165 220 180)(85 166 211 171)(86 167 212 172)(87 168 213 173)(88 169 214 174)(89 170 215 175)(90 161 216 176)
(1 127 38 224)(2 128 39 225)(3 129 40 226)(4 130 31 227)(5 121 32 228)(6 122 33 229)(7 123 34 230)(8 124 35 221)(9 125 36 222)(10 126 37 223)(11 162 206 177)(12 163 207 178)(13 164 208 179)(14 165 209 180)(15 166 210 171)(16 167 201 172)(17 168 202 173)(18 169 203 174)(19 170 204 175)(20 161 205 176)(21 193 78 181)(22 194 79 182)(23 195 80 183)(24 196 71 184)(25 197 72 185)(26 198 73 186)(27 199 74 187)(28 200 75 188)(29 191 76 189)(30 192 77 190)(41 237 98 69)(42 238 99 70)(43 239 100 61)(44 240 91 62)(45 231 92 63)(46 232 93 64)(47 233 94 65)(48 234 95 66)(49 235 96 67)(50 236 97 68)(51 105 147 140)(52 106 148 131)(53 107 149 132)(54 108 150 133)(55 109 141 134)(56 110 142 135)(57 101 143 136)(58 102 144 137)(59 103 145 138)(60 104 146 139)(81 116 217 159)(82 117 218 160)(83 118 219 151)(84 119 220 152)(85 120 211 153)(86 111 212 154)(87 112 213 155)(88 113 214 156)(89 114 215 157)(90 115 216 158)
(1 61 162)(2 62 163)(3 63 164)(4 64 165)(5 65 166)(6 66 167)(7 67 168)(8 68 169)(9 69 170)(10 70 161)(11 224 100)(12 225 91)(13 226 92)(14 227 93)(15 228 94)(16 229 95)(17 230 96)(18 221 97)(19 222 98)(20 223 99)(21 149 112)(22 150 113)(23 141 114)(24 142 115)(25 143 116)(26 144 117)(27 145 118)(28 146 119)(29 147 120)(30 148 111)(31 232 180)(32 233 171)(33 234 172)(34 235 173)(35 236 174)(36 237 175)(37 238 176)(38 239 177)(39 240 178)(40 231 179)(41 204 125)(42 205 126)(43 206 127)(44 207 128)(45 208 129)(46 209 130)(47 210 121)(48 201 122)(49 202 123)(50 203 124)(51 153 76)(52 154 77)(53 155 78)(54 156 79)(55 157 80)(56 158 71)(57 159 72)(58 160 73)(59 151 74)(60 152 75)(81 185 101)(82 186 102)(83 187 103)(84 188 104)(85 189 105)(86 190 106)(87 181 107)(88 182 108)(89 183 109)(90 184 110)(131 212 192)(132 213 193)(133 214 194)(134 215 195)(135 216 196)(136 217 197)(137 218 198)(138 219 199)(139 220 200)(140 211 191)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 51)(21 128)(22 129)(23 130)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 36)(32 37)(33 38)(34 39)(35 40)(41 119)(42 120)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(61 167)(62 168)(63 169)(64 170)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 228)(72 229)(73 230)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 151)(98 152)(99 153)(100 154)(101 212)(102 213)(103 214)(104 215)(105 216)(106 217)(107 218)(108 219)(109 220)(110 211)(141 209)(142 210)(143 201)(144 202)(145 203)(146 204)(147 205)(148 206)(149 207)(150 208)(171 238)(172 239)(173 240)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)(181 198)(182 199)(183 200)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(190 197)

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), (1,197,38,185)(2,198,39,186)(3,199,40,187)(4,200,31,188)(5,191,32,189)(6,192,33,190)(7,193,34,181)(8,194,35,182)(9,195,36,183)(10,196,37,184)(11,159,206,116)(12,160,207,117)(13,151,208,118)(14,152,209,119)(15,153,210,120)(16,154,201,111)(17,155,202,112)(18,156,203,113)(19,157,204,114)(20,158,205,115)(21,230,78,123)(22,221,79,124)(23,222,80,125)(24,223,71,126)(25,224,72,127)(26,225,73,128)(27,226,74,129)(28,227,75,130)(29,228,76,121)(30,229,77,122)(41,141,98,55)(42,142,99,56)(43,143,100,57)(44,144,91,58)(45,145,92,59)(46,146,93,60)(47,147,94,51)(48,148,95,52)(49,149,96,53)(50,150,97,54)(61,136,239,101)(62,137,240,102)(63,138,231,103)(64,139,232,104)(65,140,233,105)(66,131,234,106)(67,132,235,107)(68,133,236,108)(69,134,237,109)(70,135,238,110)(81,162,217,177)(82,163,218,178)(83,164,219,179)(84,165,220,180)(85,166,211,171)(86,167,212,172)(87,168,213,173)(88,169,214,174)(89,170,215,175)(90,161,216,176), (1,127,38,224)(2,128,39,225)(3,129,40,226)(4,130,31,227)(5,121,32,228)(6,122,33,229)(7,123,34,230)(8,124,35,221)(9,125,36,222)(10,126,37,223)(11,162,206,177)(12,163,207,178)(13,164,208,179)(14,165,209,180)(15,166,210,171)(16,167,201,172)(17,168,202,173)(18,169,203,174)(19,170,204,175)(20,161,205,176)(21,193,78,181)(22,194,79,182)(23,195,80,183)(24,196,71,184)(25,197,72,185)(26,198,73,186)(27,199,74,187)(28,200,75,188)(29,191,76,189)(30,192,77,190)(41,237,98,69)(42,238,99,70)(43,239,100,61)(44,240,91,62)(45,231,92,63)(46,232,93,64)(47,233,94,65)(48,234,95,66)(49,235,96,67)(50,236,97,68)(51,105,147,140)(52,106,148,131)(53,107,149,132)(54,108,150,133)(55,109,141,134)(56,110,142,135)(57,101,143,136)(58,102,144,137)(59,103,145,138)(60,104,146,139)(81,116,217,159)(82,117,218,160)(83,118,219,151)(84,119,220,152)(85,120,211,153)(86,111,212,154)(87,112,213,155)(88,113,214,156)(89,114,215,157)(90,115,216,158), (1,61,162)(2,62,163)(3,63,164)(4,64,165)(5,65,166)(6,66,167)(7,67,168)(8,68,169)(9,69,170)(10,70,161)(11,224,100)(12,225,91)(13,226,92)(14,227,93)(15,228,94)(16,229,95)(17,230,96)(18,221,97)(19,222,98)(20,223,99)(21,149,112)(22,150,113)(23,141,114)(24,142,115)(25,143,116)(26,144,117)(27,145,118)(28,146,119)(29,147,120)(30,148,111)(31,232,180)(32,233,171)(33,234,172)(34,235,173)(35,236,174)(36,237,175)(37,238,176)(38,239,177)(39,240,178)(40,231,179)(41,204,125)(42,205,126)(43,206,127)(44,207,128)(45,208,129)(46,209,130)(47,210,121)(48,201,122)(49,202,123)(50,203,124)(51,153,76)(52,154,77)(53,155,78)(54,156,79)(55,157,80)(56,158,71)(57,159,72)(58,160,73)(59,151,74)(60,152,75)(81,185,101)(82,186,102)(83,187,103)(84,188,104)(85,189,105)(86,190,106)(87,181,107)(88,182,108)(89,183,109)(90,184,110)(131,212,192)(132,213,193)(133,214,194)(134,215,195)(135,216,196)(136,217,197)(137,218,198)(138,219,199)(139,220,200)(140,211,191), (1,6)(2,7)(3,8)(4,9)(5,10)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(21,128)(22,129)(23,130)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,36)(32,37)(33,38)(34,39)(35,40)(41,119)(42,120)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,228)(72,229)(73,230)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,151)(98,152)(99,153)(100,154)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,211)(141,209)(142,210)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(171,238)(172,239)(173,240)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,198)(182,199)(183,200)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(190,197)>;

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), (1,197,38,185)(2,198,39,186)(3,199,40,187)(4,200,31,188)(5,191,32,189)(6,192,33,190)(7,193,34,181)(8,194,35,182)(9,195,36,183)(10,196,37,184)(11,159,206,116)(12,160,207,117)(13,151,208,118)(14,152,209,119)(15,153,210,120)(16,154,201,111)(17,155,202,112)(18,156,203,113)(19,157,204,114)(20,158,205,115)(21,230,78,123)(22,221,79,124)(23,222,80,125)(24,223,71,126)(25,224,72,127)(26,225,73,128)(27,226,74,129)(28,227,75,130)(29,228,76,121)(30,229,77,122)(41,141,98,55)(42,142,99,56)(43,143,100,57)(44,144,91,58)(45,145,92,59)(46,146,93,60)(47,147,94,51)(48,148,95,52)(49,149,96,53)(50,150,97,54)(61,136,239,101)(62,137,240,102)(63,138,231,103)(64,139,232,104)(65,140,233,105)(66,131,234,106)(67,132,235,107)(68,133,236,108)(69,134,237,109)(70,135,238,110)(81,162,217,177)(82,163,218,178)(83,164,219,179)(84,165,220,180)(85,166,211,171)(86,167,212,172)(87,168,213,173)(88,169,214,174)(89,170,215,175)(90,161,216,176), (1,127,38,224)(2,128,39,225)(3,129,40,226)(4,130,31,227)(5,121,32,228)(6,122,33,229)(7,123,34,230)(8,124,35,221)(9,125,36,222)(10,126,37,223)(11,162,206,177)(12,163,207,178)(13,164,208,179)(14,165,209,180)(15,166,210,171)(16,167,201,172)(17,168,202,173)(18,169,203,174)(19,170,204,175)(20,161,205,176)(21,193,78,181)(22,194,79,182)(23,195,80,183)(24,196,71,184)(25,197,72,185)(26,198,73,186)(27,199,74,187)(28,200,75,188)(29,191,76,189)(30,192,77,190)(41,237,98,69)(42,238,99,70)(43,239,100,61)(44,240,91,62)(45,231,92,63)(46,232,93,64)(47,233,94,65)(48,234,95,66)(49,235,96,67)(50,236,97,68)(51,105,147,140)(52,106,148,131)(53,107,149,132)(54,108,150,133)(55,109,141,134)(56,110,142,135)(57,101,143,136)(58,102,144,137)(59,103,145,138)(60,104,146,139)(81,116,217,159)(82,117,218,160)(83,118,219,151)(84,119,220,152)(85,120,211,153)(86,111,212,154)(87,112,213,155)(88,113,214,156)(89,114,215,157)(90,115,216,158), (1,61,162)(2,62,163)(3,63,164)(4,64,165)(5,65,166)(6,66,167)(7,67,168)(8,68,169)(9,69,170)(10,70,161)(11,224,100)(12,225,91)(13,226,92)(14,227,93)(15,228,94)(16,229,95)(17,230,96)(18,221,97)(19,222,98)(20,223,99)(21,149,112)(22,150,113)(23,141,114)(24,142,115)(25,143,116)(26,144,117)(27,145,118)(28,146,119)(29,147,120)(30,148,111)(31,232,180)(32,233,171)(33,234,172)(34,235,173)(35,236,174)(36,237,175)(37,238,176)(38,239,177)(39,240,178)(40,231,179)(41,204,125)(42,205,126)(43,206,127)(44,207,128)(45,208,129)(46,209,130)(47,210,121)(48,201,122)(49,202,123)(50,203,124)(51,153,76)(52,154,77)(53,155,78)(54,156,79)(55,157,80)(56,158,71)(57,159,72)(58,160,73)(59,151,74)(60,152,75)(81,185,101)(82,186,102)(83,187,103)(84,188,104)(85,189,105)(86,190,106)(87,181,107)(88,182,108)(89,183,109)(90,184,110)(131,212,192)(132,213,193)(133,214,194)(134,215,195)(135,216,196)(136,217,197)(137,218,198)(138,219,199)(139,220,200)(140,211,191), (1,6)(2,7)(3,8)(4,9)(5,10)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(21,128)(22,129)(23,130)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,36)(32,37)(33,38)(34,39)(35,40)(41,119)(42,120)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,228)(72,229)(73,230)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,151)(98,152)(99,153)(100,154)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,211)(141,209)(142,210)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(171,238)(172,239)(173,240)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,198)(182,199)(183,200)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(190,197) );

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)], [(1,197,38,185),(2,198,39,186),(3,199,40,187),(4,200,31,188),(5,191,32,189),(6,192,33,190),(7,193,34,181),(8,194,35,182),(9,195,36,183),(10,196,37,184),(11,159,206,116),(12,160,207,117),(13,151,208,118),(14,152,209,119),(15,153,210,120),(16,154,201,111),(17,155,202,112),(18,156,203,113),(19,157,204,114),(20,158,205,115),(21,230,78,123),(22,221,79,124),(23,222,80,125),(24,223,71,126),(25,224,72,127),(26,225,73,128),(27,226,74,129),(28,227,75,130),(29,228,76,121),(30,229,77,122),(41,141,98,55),(42,142,99,56),(43,143,100,57),(44,144,91,58),(45,145,92,59),(46,146,93,60),(47,147,94,51),(48,148,95,52),(49,149,96,53),(50,150,97,54),(61,136,239,101),(62,137,240,102),(63,138,231,103),(64,139,232,104),(65,140,233,105),(66,131,234,106),(67,132,235,107),(68,133,236,108),(69,134,237,109),(70,135,238,110),(81,162,217,177),(82,163,218,178),(83,164,219,179),(84,165,220,180),(85,166,211,171),(86,167,212,172),(87,168,213,173),(88,169,214,174),(89,170,215,175),(90,161,216,176)], [(1,127,38,224),(2,128,39,225),(3,129,40,226),(4,130,31,227),(5,121,32,228),(6,122,33,229),(7,123,34,230),(8,124,35,221),(9,125,36,222),(10,126,37,223),(11,162,206,177),(12,163,207,178),(13,164,208,179),(14,165,209,180),(15,166,210,171),(16,167,201,172),(17,168,202,173),(18,169,203,174),(19,170,204,175),(20,161,205,176),(21,193,78,181),(22,194,79,182),(23,195,80,183),(24,196,71,184),(25,197,72,185),(26,198,73,186),(27,199,74,187),(28,200,75,188),(29,191,76,189),(30,192,77,190),(41,237,98,69),(42,238,99,70),(43,239,100,61),(44,240,91,62),(45,231,92,63),(46,232,93,64),(47,233,94,65),(48,234,95,66),(49,235,96,67),(50,236,97,68),(51,105,147,140),(52,106,148,131),(53,107,149,132),(54,108,150,133),(55,109,141,134),(56,110,142,135),(57,101,143,136),(58,102,144,137),(59,103,145,138),(60,104,146,139),(81,116,217,159),(82,117,218,160),(83,118,219,151),(84,119,220,152),(85,120,211,153),(86,111,212,154),(87,112,213,155),(88,113,214,156),(89,114,215,157),(90,115,216,158)], [(1,61,162),(2,62,163),(3,63,164),(4,64,165),(5,65,166),(6,66,167),(7,67,168),(8,68,169),(9,69,170),(10,70,161),(11,224,100),(12,225,91),(13,226,92),(14,227,93),(15,228,94),(16,229,95),(17,230,96),(18,221,97),(19,222,98),(20,223,99),(21,149,112),(22,150,113),(23,141,114),(24,142,115),(25,143,116),(26,144,117),(27,145,118),(28,146,119),(29,147,120),(30,148,111),(31,232,180),(32,233,171),(33,234,172),(34,235,173),(35,236,174),(36,237,175),(37,238,176),(38,239,177),(39,240,178),(40,231,179),(41,204,125),(42,205,126),(43,206,127),(44,207,128),(45,208,129),(46,209,130),(47,210,121),(48,201,122),(49,202,123),(50,203,124),(51,153,76),(52,154,77),(53,155,78),(54,156,79),(55,157,80),(56,158,71),(57,159,72),(58,160,73),(59,151,74),(60,152,75),(81,185,101),(82,186,102),(83,187,103),(84,188,104),(85,189,105),(86,190,106),(87,181,107),(88,182,108),(89,183,109),(90,184,110),(131,212,192),(132,213,193),(133,214,194),(134,215,195),(135,216,196),(136,217,197),(137,218,198),(138,219,199),(139,220,200),(140,211,191)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,51),(21,128),(22,129),(23,130),(24,121),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,36),(32,37),(33,38),(34,39),(35,40),(41,119),(42,120),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(61,167),(62,168),(63,169),(64,170),(65,161),(66,162),(67,163),(68,164),(69,165),(70,166),(71,228),(72,229),(73,230),(74,221),(75,222),(76,223),(77,224),(78,225),(79,226),(80,227),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,151),(98,152),(99,153),(100,154),(101,212),(102,213),(103,214),(104,215),(105,216),(106,217),(107,218),(108,219),(109,220),(110,211),(141,209),(142,210),(143,201),(144,202),(145,203),(146,204),(147,205),(148,206),(149,207),(150,208),(171,238),(172,239),(173,240),(174,231),(175,232),(176,233),(177,234),(178,235),(179,236),(180,237),(181,198),(182,199),(183,200),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(190,197)]])

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 30A ··· 30L 40A ··· 40P 60A ··· 60X order 1 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 12 12 2 2 2 4 4 1 1 1 1 2 2 2 6 6 6 6 1 ··· 1 12 ··· 12 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 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 SD16 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×SD16 C5×C3⋊D4 C5×C3⋊D4 Q8⋊2S3 C5×Q8⋊2S3 kernel C10×Q8⋊2S3 C10×C3⋊C8 C5×Q8⋊2S3 C10×D12 Q8×C30 C2×Q8⋊2S3 C2×C3⋊C8 Q8⋊2S3 C2×D12 C6×Q8 Q8×C10 C60 C2×C30 C2×C20 C5×Q8 C30 C20 C2×C10 C2×Q8 C12 C2×C6 C2×C4 Q8 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×Q82S3 in GL4(𝔽241) generated by

 150 0 0 0 0 150 0 0 0 0 154 0 0 0 0 154
,
 240 0 0 0 0 240 0 0 0 0 240 3 0 0 160 1
,
 70 140 0 0 101 171 0 0 0 0 0 57 0 0 93 0
,
 0 1 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 240 0 0 0 160 1
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,154,0,0,0,0,154],[240,0,0,0,0,240,0,0,0,0,240,160,0,0,3,1],[70,101,0,0,140,171,0,0,0,0,0,93,0,0,57,0],[0,240,0,0,1,240,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,240,160,0,0,0,1] >;

C10×Q82S3 in GAP, Magma, Sage, TeX

C_{10}\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C10xQ8:2S3");
// GroupNames label

G:=SmallGroup(480,820);
// by ID

G=gap.SmallGroup(480,820);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,436,4204,1068,102,15686]);
// 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,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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