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## G = C5×C23.16D6order 480 = 25·3·5

### Direct product of C5 and C23.16D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C23.16D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C2×C10 — C5×C23.16D6
 Lower central C3 — C6 — C5×C23.16D6
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×C23.16D6
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, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 276 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3, C5×Dic3, C60, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C23.16D6, C10×Dic3, C10×Dic3, C2×C60, C22×C30, C5×C42⋊C2, Dic3×C20, C5×Dic3⋊C4, C5×C6.D4, C15×C22⋊C4, Dic3×C2×C10, C5×C23.16D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C42⋊C2, C2×C20, C22×C10, S3×C2×C4, D42S3, S3×C10, C22×C20, C5×C4○D4, C23.16D6, S3×C20, S3×C2×C10, C5×C42⋊C2, S3×C2×C20, C5×D42S3, C5×C23.16D6

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

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M ··· 10T 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 20AG ··· 20BD 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 6 ··· 6 1 1 1 1 2 2 2 4 4 1 ··· 1 2 ··· 2 4 4 4 4 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4 4 ··· 4

150 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 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 S3 D6 D6 C4○D4 C4×S3 C5×S3 S3×C10 S3×C10 C5×C4○D4 S3×C20 D4⋊2S3 C5×D4⋊2S3 kernel C5×C23.16D6 Dic3×C20 C5×Dic3⋊C4 C5×C6.D4 C15×C22⋊C4 Dic3×C2×C10 C10×Dic3 C23.16D6 C4×Dic3 Dic3⋊C4 C6.D4 C3×C22⋊C4 C22×Dic3 C2×Dic3 C5×C22⋊C4 C2×C20 C22×C10 C30 C2×C10 C22⋊C4 C2×C4 C23 C6 C22 C10 C2 # reps 1 2 2 1 1 1 8 4 8 8 4 4 4 32 1 2 1 4 4 4 8 4 16 16 2 8

Matrix representation of C5×C23.16D6 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 9 0 0 0 0 9
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 50 1
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 0 11 0 0 50 50 0 0 0 0 11 59 0 0 0 50
,
 27 31 0 0 4 34 0 0 0 0 60 39 0 0 0 1
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,60,50,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[0,50,0,0,11,50,0,0,0,0,11,0,0,0,59,50],[27,4,0,0,31,34,0,0,0,0,60,0,0,0,39,1] >;

C5×C23.16D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{16}D_6
% in TeX

G:=Group("C5xC2^3.16D6");
// GroupNames label

G:=SmallGroup(480,756);
// by ID

G=gap.SmallGroup(480,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,891,226,15686]);
// 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,b*c=c*b,e*b*e^-1=f*b*f^-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=e^5>;
// generators/relations

׿
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