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G = C5×C23.23D6order 480 = 25·3·5

Direct product of C5 and C23.23D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.23D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C2×C10 — C5×C23.23D6
 Lower central C3 — C2×C6 — C5×C23.23D6
 Upper central C1 — C2×C10 — D4×C10

Generators and relations for C5×C23.23D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=e6=1, 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=ce-1 >

Subgroups: 324 in 156 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C30, C22.D4, C2×C20, C2×C20, C5×D4, C22×C10, Dic3⋊C4, C6.D4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C23.23D6, C10×Dic3, C10×Dic3, C2×C60, D4×C15, C22×C30, C5×C22.D4, C5×Dic3⋊C4, C5×C6.D4, C5×C6.D4, Dic3×C2×C10, D4×C30, C5×C23.23D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C22.D4, C5×D4, C22×C10, D42S3, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C23.23D6, C5×C3⋊D4, S3×C2×C10, C5×C22.D4, C5×D42S3, C10×C3⋊D4, C5×C23.23D6

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20T 20U ··· 20AB 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 10 10 10 10 12 12 15 15 15 15 20 20 20 20 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 4 2 4 6 6 6 6 12 12 1 1 1 1 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 6 ··· 6 12 ··· 12 2 ··· 2 4 ··· 4 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 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D6 D6 C4○D4 C3⋊D4 C5×S3 C5×D4 S3×C10 S3×C10 C5×C4○D4 C5×C3⋊D4 D4⋊2S3 C5×D4⋊2S3 kernel C5×C23.23D6 C5×Dic3⋊C4 C5×C6.D4 Dic3×C2×C10 D4×C30 C23.23D6 Dic3⋊C4 C6.D4 C22×Dic3 C6×D4 D4×C10 C2×C30 C2×C20 C22×C10 C30 C2×C10 C2×D4 C2×C6 C2×C4 C23 C6 C22 C10 C2 # reps 1 2 3 1 1 4 8 12 4 4 1 2 1 2 4 4 4 8 4 8 16 16 2 8

Matrix representation of C5×C23.23D6 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 1 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 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
,
 1 59 0 0 0 60 0 0 0 0 9 9 0 0 52 18
,
 11 39 0 0 11 50 0 0 0 0 38 21 0 0 59 23
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,1,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,59,60,0,0,0,0,9,52,0,0,9,18],[11,11,0,0,39,50,0,0,0,0,38,59,0,0,21,23] >;

C5×C23.23D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{23}D_6
% in TeX

G:=Group("C5xC2^3.23D6");
// GroupNames label

G:=SmallGroup(480,814);
// by ID

G=gap.SmallGroup(480,814);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,891,15686]);
// Polycyclic

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

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