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## G = C10×Q8⋊3S3order 480 = 25·3·5

### Direct product of C10 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×Q8⋊3S3
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C2×C20 — C10×Q8⋊3S3
 Lower central C3 — C6 — C10×Q8⋊3S3
 Upper central C1 — C2×C10 — Q8×C10

Generators and relations for C10×Q83S3
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 >

Subgroups: 676 in 328 conjugacy classes, 178 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, S3×C2×C4, C2×D12, Q83S3, C6×Q8, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×Q83S3, S3×C20, C5×D12, C10×Dic3, C2×C60, Q8×C15, S3×C2×C10, C10×C4○D4, S3×C2×C20, C10×D12, C5×Q83S3, Q8×C30, C10×Q83S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C24, C2×C10, C22×S3, C5×S3, C2×C4○D4, C22×C10, Q83S3, S3×C23, S3×C10, C5×C4○D4, C23×C10, C2×Q83S3, S3×C2×C10, C10×C4○D4, C5×Q83S3, S3×C22×C10, C10×Q83S3

Smallest permutation representation of C10×Q83S3
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)
(1 213 207 173)(2 214 208 174)(3 215 209 175)(4 216 210 176)(5 217 201 177)(6 218 202 178)(7 219 203 179)(8 220 204 180)(9 211 205 171)(10 212 206 172)(11 62 194 147)(12 63 195 148)(13 64 196 149)(14 65 197 150)(15 66 198 141)(16 67 199 142)(17 68 200 143)(18 69 191 144)(19 70 192 145)(20 61 193 146)(21 188 31 236)(22 189 32 237)(23 190 33 238)(24 181 34 239)(25 182 35 240)(26 183 36 231)(27 184 37 232)(28 185 38 233)(29 186 39 234)(30 187 40 235)(41 157 105 113)(42 158 106 114)(43 159 107 115)(44 160 108 116)(45 151 109 117)(46 152 110 118)(47 153 101 119)(48 154 102 120)(49 155 103 111)(50 156 104 112)(51 224 92 168)(52 225 93 169)(53 226 94 170)(54 227 95 161)(55 228 96 162)(56 229 97 163)(57 230 98 164)(58 221 99 165)(59 222 100 166)(60 223 91 167)(71 124 136 89)(72 125 137 90)(73 126 138 81)(74 127 139 82)(75 128 140 83)(76 129 131 84)(77 130 132 85)(78 121 133 86)(79 122 134 87)(80 123 135 88)
(1 237 207 189)(2 238 208 190)(3 239 209 181)(4 240 210 182)(5 231 201 183)(6 232 202 184)(7 233 203 185)(8 234 204 186)(9 235 205 187)(10 236 206 188)(11 230 194 164)(12 221 195 165)(13 222 196 166)(14 223 197 167)(15 224 198 168)(16 225 199 169)(17 226 200 170)(18 227 191 161)(19 228 192 162)(20 229 193 163)(21 212 31 172)(22 213 32 173)(23 214 33 174)(24 215 34 175)(25 216 35 176)(26 217 36 177)(27 218 37 178)(28 219 38 179)(29 220 39 180)(30 211 40 171)(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)(51 141 92 66)(52 142 93 67)(53 143 94 68)(54 144 95 69)(55 145 96 70)(56 146 97 61)(57 147 98 62)(58 148 99 63)(59 149 100 64)(60 150 91 65)(81 159 126 115)(82 160 127 116)(83 151 128 117)(84 152 129 118)(85 153 130 119)(86 154 121 120)(87 155 122 111)(88 156 123 112)(89 157 124 113)(90 158 125 114)
(1 157 97)(2 158 98)(3 159 99)(4 160 100)(5 151 91)(6 152 92)(7 153 93)(8 154 94)(9 155 95)(10 156 96)(11 33 72)(12 34 73)(13 35 74)(14 36 75)(15 37 76)(16 38 77)(17 39 78)(18 40 79)(19 31 80)(20 32 71)(21 135 192)(22 136 193)(23 137 194)(24 138 195)(25 139 196)(26 140 197)(27 131 198)(28 132 199)(29 133 200)(30 134 191)(41 229 173)(42 230 174)(43 221 175)(44 222 176)(45 223 177)(46 224 178)(47 225 179)(48 226 180)(49 227 171)(50 228 172)(51 202 118)(52 203 119)(53 204 120)(54 205 111)(55 206 112)(56 207 113)(57 208 114)(58 209 115)(59 210 116)(60 201 117)(61 237 124)(62 238 125)(63 239 126)(64 240 127)(65 231 128)(66 232 129)(67 233 130)(68 234 121)(69 235 122)(70 236 123)(81 148 181)(82 149 182)(83 150 183)(84 141 184)(85 142 185)(86 143 186)(87 144 187)(88 145 188)(89 146 189)(90 147 190)(101 169 219)(102 170 220)(103 161 211)(104 162 212)(105 163 213)(106 164 214)(107 165 215)(108 166 216)(109 167 217)(110 168 218)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 31)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 206)(22 207)(23 208)(24 209)(25 210)(26 201)(27 202)(28 203)(29 204)(30 205)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(71 97)(72 98)(73 99)(74 100)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 165)(82 166)(83 167)(84 168)(85 169)(86 170)(87 161)(88 162)(89 163)(90 164)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 141)(111 191)(112 192)(113 193)(114 194)(115 195)(116 196)(117 197)(118 198)(119 199)(120 200)(121 226)(122 227)(123 228)(124 229)(125 230)(126 221)(127 222)(128 223)(129 224)(130 225)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 231)(178 232)(179 233)(180 234)(181 215)(182 216)(183 217)(184 218)(185 219)(186 220)(187 211)(188 212)(189 213)(190 214)

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

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

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

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

Matrix representation of C10×Q83S3 in 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] >;

C10×Q83S3 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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