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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 [×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], Q83S3 [×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×Q83S3, 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×Q83S3 [×8], Q8×C30, C10×Q83S3
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], Q83S3 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×Q83S3, S3×C2×C10 [×7], C10×C4○D4, C5×Q83S3 [×2], 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)
(1 117 207 173)(2 118 208 174)(3 119 209 175)(4 120 210 176)(5 111 201 177)(6 112 202 178)(7 113 203 179)(8 114 204 180)(9 115 205 171)(10 116 206 172)(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 82 54 236)(22 83 55 237)(23 84 56 238)(24 85 57 239)(25 86 58 240)(26 87 59 231)(27 88 60 232)(28 89 51 233)(29 90 52 234)(30 81 53 235)(31 229 97 163)(32 230 98 164)(33 221 99 165)(34 222 100 166)(35 223 91 167)(36 224 92 168)(37 225 93 169)(38 226 94 170)(39 227 95 161)(40 228 96 162)(41 155 105 219)(42 156 106 220)(43 157 107 211)(44 158 108 212)(45 159 109 213)(46 160 110 214)(47 151 101 215)(48 152 102 216)(49 153 103 217)(50 154 104 218)(71 124 136 185)(72 125 137 186)(73 126 138 187)(74 127 139 188)(75 128 140 189)(76 129 131 190)(77 130 132 181)(78 121 133 182)(79 122 134 183)(80 123 135 184)
(1 237 207 83)(2 238 208 84)(3 239 209 85)(4 240 210 86)(5 231 201 87)(6 232 202 88)(7 233 203 89)(8 234 204 90)(9 235 205 81)(10 236 206 82)(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)(21 116 54 172)(22 117 55 173)(23 118 56 174)(24 119 57 175)(25 120 58 176)(26 111 59 177)(27 112 60 178)(28 113 51 179)(29 114 52 180)(30 115 53 171)(31 146 97 61)(32 147 98 62)(33 148 99 63)(34 149 100 64)(35 150 91 65)(36 141 92 66)(37 142 93 67)(38 143 94 68)(39 144 95 69)(40 145 96 70)(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)(121 216 182 152)(122 217 183 153)(123 218 184 154)(124 219 185 155)(125 220 186 156)(126 211 187 157)(127 212 188 158)(128 213 189 159)(129 214 190 160)(130 215 181 151)
(1 155 97)(2 156 98)(3 157 99)(4 158 100)(5 159 91)(6 160 92)(7 151 93)(8 152 94)(9 153 95)(10 154 96)(11 56 72)(12 57 73)(13 58 74)(14 59 75)(15 60 76)(16 51 77)(17 52 78)(18 53 79)(19 54 80)(20 55 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)(31 207 219)(32 208 220)(33 209 211)(34 210 212)(35 201 213)(36 202 214)(37 203 215)(38 204 216)(39 205 217)(40 206 218)(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)(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)(81 183 144)(82 184 145)(83 185 146)(84 186 147)(85 187 148)(86 188 149)(87 189 150)(88 190 141)(89 181 142)(90 182 143)(101 169 113)(102 170 114)(103 161 115)(104 162 116)(105 163 117)(106 164 118)(107 165 119)(108 166 120)(109 167 111)(110 168 112)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 51)(8 52)(9 53)(10 54)(11 156)(12 157)(13 158)(14 159)(15 160)(16 151)(17 152)(18 153)(19 154)(20 155)(21 206)(22 207)(23 208)(24 209)(25 210)(26 201)(27 202)(28 203)(29 204)(30 205)(31 136)(32 137)(33 138)(34 139)(35 140)(36 131)(37 132)(38 133)(39 134)(40 135)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(71 97)(72 98)(73 99)(74 100)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 111)(88 112)(89 113)(90 114)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 141)(121 226)(122 227)(123 228)(124 229)(125 230)(126 221)(127 222)(128 223)(129 224)(130 225)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 181)(170 182)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 231)(178 232)(179 233)(180 234)(191 217)(192 218)(193 219)(194 220)(195 211)(196 212)(197 213)(198 214)(199 215)(200 216)

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

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

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

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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