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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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C10, C10 [×2], C10 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C22×S3, C5×S3 [×2], C30, C30 [×2], C2×SD16, C40 [×2], C2×C20, C2×C20, C5×D4 [×3], C5×Q8 [×2], C5×Q8, C22×C10, C2×C3⋊C8, Q82S3 [×4], C2×D12, C6×Q8, C60 [×2], C60 [×2], S3×C10 [×4], C2×C30, C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C2×Q82S3, C5×C3⋊C8 [×2], C5×D12 [×2], C5×D12, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, S3×C2×C10, C10×SD16, C10×C3⋊C8, C5×Q82S3 [×4], C10×D12, Q8×C30, C10×Q82S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], SD16 [×2], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C2×SD16, C5×D4 [×2], C22×C10, Q82S3 [×2], C2×C3⋊D4, S3×C10 [×3], C5×SD16 [×2], D4×C10, C2×Q82S3, C5×C3⋊D4 [×2], S3×C2×C10, C10×SD16, C5×Q82S3 [×2], 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 77 14 219)(2 78 15 220)(3 79 16 211)(4 80 17 212)(5 71 18 213)(6 72 19 214)(7 73 20 215)(8 74 11 216)(9 75 12 217)(10 76 13 218)(21 222 118 135)(22 223 119 136)(23 224 120 137)(24 225 111 138)(25 226 112 139)(26 227 113 140)(27 228 114 131)(28 229 115 132)(29 230 116 133)(30 221 117 134)(31 143 57 210)(32 144 58 201)(33 145 59 202)(34 146 60 203)(35 147 51 204)(36 148 52 205)(37 149 53 206)(38 150 54 207)(39 141 55 208)(40 142 56 209)(41 127 108 61)(42 128 109 62)(43 129 110 63)(44 130 101 64)(45 121 102 65)(46 122 103 66)(47 123 104 67)(48 124 105 68)(49 125 106 69)(50 126 107 70)(81 160 179 197)(82 151 180 198)(83 152 171 199)(84 153 172 200)(85 154 173 191)(86 155 174 192)(87 156 175 193)(88 157 176 194)(89 158 177 195)(90 159 178 196)(91 170 237 181)(92 161 238 182)(93 162 239 183)(94 163 240 184)(95 164 231 185)(96 165 232 186)(97 166 233 187)(98 167 234 188)(99 168 235 189)(100 169 236 190)
(1 45 14 102)(2 46 15 103)(3 47 16 104)(4 48 17 105)(5 49 18 106)(6 50 19 107)(7 41 20 108)(8 42 11 109)(9 43 12 110)(10 44 13 101)(21 85 118 173)(22 86 119 174)(23 87 120 175)(24 88 111 176)(25 89 112 177)(26 90 113 178)(27 81 114 179)(28 82 115 180)(29 83 116 171)(30 84 117 172)(31 239 57 93)(32 240 58 94)(33 231 59 95)(34 232 60 96)(35 233 51 97)(36 234 52 98)(37 235 53 99)(38 236 54 100)(39 237 55 91)(40 238 56 92)(61 215 127 73)(62 216 128 74)(63 217 129 75)(64 218 130 76)(65 219 121 77)(66 220 122 78)(67 211 123 79)(68 212 124 80)(69 213 125 71)(70 214 126 72)(131 160 228 197)(132 151 229 198)(133 152 230 199)(134 153 221 200)(135 154 222 191)(136 155 223 192)(137 156 224 193)(138 157 225 194)(139 158 226 195)(140 159 227 196)(141 170 208 181)(142 161 209 182)(143 162 210 183)(144 163 201 184)(145 164 202 185)(146 165 203 186)(147 166 204 187)(148 167 205 188)(149 168 206 189)(150 169 207 190)
(1 93 160)(2 94 151)(3 95 152)(4 96 153)(5 97 154)(6 98 155)(7 99 156)(8 100 157)(9 91 158)(10 92 159)(11 236 194)(12 237 195)(13 238 196)(14 239 197)(15 240 198)(16 231 199)(17 232 200)(18 233 191)(19 234 192)(20 235 193)(21 69 204)(22 70 205)(23 61 206)(24 62 207)(25 63 208)(26 64 209)(27 65 210)(28 66 201)(29 67 202)(30 68 203)(31 228 45)(32 229 46)(33 230 47)(34 221 48)(35 222 49)(36 223 50)(37 224 41)(38 225 42)(39 226 43)(40 227 44)(51 135 106)(52 136 107)(53 137 108)(54 138 109)(55 139 110)(56 140 101)(57 131 102)(58 132 103)(59 133 104)(60 134 105)(71 166 173)(72 167 174)(73 168 175)(74 169 176)(75 170 177)(76 161 178)(77 162 179)(78 163 180)(79 164 171)(80 165 172)(81 219 183)(82 220 184)(83 211 185)(84 212 186)(85 213 187)(86 214 188)(87 215 189)(88 216 190)(89 217 181)(90 218 182)(111 128 150)(112 129 141)(113 130 142)(114 121 143)(115 122 144)(116 123 145)(117 124 146)(118 125 147)(119 126 148)(120 127 149)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 119)(32 120)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 121)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 101)(70 102)(71 218)(72 219)(73 220)(74 211)(75 212)(76 213)(77 214)(78 215)(79 216)(80 217)(81 167)(82 168)(83 169)(84 170)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 151)(100 152)(131 205)(132 206)(133 207)(134 208)(135 209)(136 210)(137 201)(138 202)(139 203)(140 204)(141 221)(142 222)(143 223)(144 224)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(171 190)(172 181)(173 182)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)(191 238)(192 239)(193 240)(194 231)(195 232)(196 233)(197 234)(198 235)(199 236)(200 237)

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

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

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

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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