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## G = C5×C23.11D6order 480 = 25·3·5

### Direct product of C5 and C23.11D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.11D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C5×D6⋊C4 — C5×C23.11D6
 Lower central C3 — C2×C6 — C5×C23.11D6
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×C23.11D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 372 in 152 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, C20, C2×C10, C2×C10, Dic6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4.4D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C22⋊C4, D4×C10, Q8×C10, C23.11D6, C5×Dic6, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, C5×C4.4D4, Dic3×C20, C5×D6⋊C4, C5×C6.D4, C15×C22⋊C4, C10×Dic6, C10×C3⋊D4, C5×C23.11D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C22×S3, C5×S3, C4.4D4, C5×D4, C22×C10, C4○D12, S3×D4, D42S3, S3×C10, D4×C10, C5×C4○D4, C23.11D6, S3×C2×C10, C5×C4.4D4, C5×C4○D12, C5×S3×D4, C5×D42S3, C5×C23.11D6

Smallest permutation representation of C5×C23.11D6
On 240 points
Generators in S240
(1 160 237 119 213)(2 161 238 120 214)(3 162 239 109 215)(4 163 240 110 216)(5 164 229 111 205)(6 165 230 112 206)(7 166 231 113 207)(8 167 232 114 208)(9 168 233 115 209)(10 157 234 116 210)(11 158 235 117 211)(12 159 236 118 212)(13 59 100 140 77)(14 60 101 141 78)(15 49 102 142 79)(16 50 103 143 80)(17 51 104 144 81)(18 52 105 133 82)(19 53 106 134 83)(20 54 107 135 84)(21 55 108 136 73)(22 56 97 137 74)(23 57 98 138 75)(24 58 99 139 76)(25 42 95 170 67)(26 43 96 171 68)(27 44 85 172 69)(28 45 86 173 70)(29 46 87 174 71)(30 47 88 175 72)(31 48 89 176 61)(32 37 90 177 62)(33 38 91 178 63)(34 39 92 179 64)(35 40 93 180 65)(36 41 94 169 66)(121 182 153 200 226)(122 183 154 201 227)(123 184 155 202 228)(124 185 156 203 217)(125 186 145 204 218)(126 187 146 193 219)(127 188 147 194 220)(128 189 148 195 221)(129 190 149 196 222)(130 191 150 197 223)(131 192 151 198 224)(132 181 152 199 225)
(1 193)(2 17)(3 195)(4 19)(5 197)(6 21)(7 199)(8 23)(9 201)(10 13)(11 203)(12 15)(14 39)(16 41)(18 43)(20 45)(22 47)(24 37)(25 147)(26 82)(27 149)(28 84)(29 151)(30 74)(31 153)(32 76)(33 155)(34 78)(35 145)(36 80)(38 202)(40 204)(42 194)(44 196)(46 198)(48 200)(49 159)(50 94)(51 161)(52 96)(53 163)(54 86)(55 165)(56 88)(57 167)(58 90)(59 157)(60 92)(61 182)(62 139)(63 184)(64 141)(65 186)(66 143)(67 188)(68 133)(69 190)(70 135)(71 192)(72 137)(73 206)(75 208)(77 210)(79 212)(81 214)(83 216)(85 222)(87 224)(89 226)(91 228)(93 218)(95 220)(97 175)(98 232)(99 177)(100 234)(101 179)(102 236)(103 169)(104 238)(105 171)(106 240)(107 173)(108 230)(109 189)(110 134)(111 191)(112 136)(113 181)(114 138)(115 183)(116 140)(117 185)(118 142)(119 187)(120 144)(121 176)(122 233)(123 178)(124 235)(125 180)(126 237)(127 170)(128 239)(129 172)(130 229)(131 174)(132 231)(146 213)(148 215)(150 205)(152 207)(154 209)(156 211)(158 217)(160 219)(162 221)(164 223)(166 225)(168 227)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 202)(14 203)(15 204)(16 193)(17 194)(18 195)(19 196)(20 197)(21 198)(22 199)(23 200)(24 201)(25 214)(26 215)(27 216)(28 205)(29 206)(30 207)(31 208)(32 209)(33 210)(34 211)(35 212)(36 213)(49 218)(50 219)(51 220)(52 221)(53 222)(54 223)(55 224)(56 225)(57 226)(58 227)(59 228)(60 217)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 109)(69 110)(70 111)(71 112)(72 113)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 132)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(133 189)(134 190)(135 191)(136 192)(137 181)(138 182)(139 183)(140 184)(141 185)(142 186)(143 187)(144 188)(169 237)(170 238)(171 239)(172 240)(173 229)(174 230)(175 231)(176 232)(177 233)(178 234)(179 235)(180 236)
(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 6 47 40)(2 39 48 5)(3 4 37 38)(7 12 41 46)(8 45 42 11)(9 10 43 44)(13 24 196 195)(14 194 197 23)(15 22 198 193)(16 204 199 21)(17 20 200 203)(18 202 201 19)(25 211 208 28)(26 27 209 210)(29 207 212 36)(30 35 213 206)(31 205 214 34)(32 33 215 216)(49 56 224 219)(50 218 225 55)(51 54 226 217)(52 228 227 53)(57 60 220 223)(58 222 221 59)(61 111 120 64)(62 63 109 110)(65 119 112 72)(66 71 113 118)(67 117 114 70)(68 69 115 116)(73 80 145 152)(74 151 146 79)(75 78 147 150)(76 149 148 77)(81 84 153 156)(82 155 154 83)(85 168 157 96)(86 95 158 167)(87 166 159 94)(88 93 160 165)(89 164 161 92)(90 91 162 163)(97 131 126 102)(98 101 127 130)(99 129 128 100)(103 125 132 108)(104 107 121 124)(105 123 122 106)(133 184 183 134)(135 182 185 144)(136 143 186 181)(137 192 187 142)(138 141 188 191)(139 190 189 140)(169 174 231 236)(170 235 232 173)(171 172 233 234)(175 180 237 230)(176 229 238 179)(177 178 239 240)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M 10N 10O 10P 10Q 10R 10S 10T 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20AB 20AC 20AD 20AE 20AF 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 12 2 2 2 4 6 6 6 6 12 1 1 1 1 2 2 2 4 4 1 ··· 1 4 4 4 4 12 12 12 12 4 4 4 4 2 2 2 2 2 ··· 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 S3 D4 D6 D6 C4○D4 C5×S3 C5×D4 C4○D12 S3×C10 S3×C10 C5×C4○D4 C5×C4○D12 S3×D4 D4⋊2S3 C5×S3×D4 C5×D4⋊2S3 kernel C5×C23.11D6 Dic3×C20 C5×D6⋊C4 C5×C6.D4 C15×C22⋊C4 C10×Dic6 C10×C3⋊D4 C23.11D6 C4×Dic3 D6⋊C4 C6.D4 C3×C22⋊C4 C2×Dic6 C2×C3⋊D4 C5×C22⋊C4 C5×Dic3 C2×C20 C22×C10 C30 C22⋊C4 Dic3 C10 C2×C4 C23 C6 C2 C10 C10 C2 C2 # reps 1 1 2 1 1 1 1 4 4 8 4 4 4 4 1 2 2 1 4 4 8 4 8 4 16 16 1 1 4 4

Matrix representation of C5×C23.11D6 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 20 0 0 0 0 20
,
 1 59 0 0 0 60 0 0 0 0 9 18 0 0 43 52
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 50 22 0 0 50 11 0 0 0 0 0 11 0 0 50 50
,
 11 39 0 0 0 50 0 0 0 0 0 11 0 0 11 0
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,20,0,0,0,0,20],[1,0,0,0,59,60,0,0,0,0,9,43,0,0,18,52],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[50,50,0,0,22,11,0,0,0,0,0,50,0,0,11,50],[11,0,0,0,39,50,0,0,0,0,0,11,0,0,11,0] >;

C5×C23.11D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{11}D_6
% in TeX

G:=Group("C5xC2^3.11D6");
// GroupNames label

G:=SmallGroup(480,764);
// by ID

G=gap.SmallGroup(480,764);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,288,2606,891,15686]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations

׿
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