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## G = C5×C23.8D6order 480 = 25·3·5

### Direct product of C5 and C23.8D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.8D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C20 — C5×C23.8D6
 Lower central C3 — C2×C6 — C5×C23.8D6
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×C23.8D6
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=e5 >

Subgroups: 244 in 120 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×C6, C30, C30, C422C2, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C5×Dic3, C60, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C23.8D6, C10×Dic3, C2×C60, C22×C30, C5×C422C2, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×C6.D4, C15×C22⋊C4, C5×C23.8D6
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C422C2, C22×C10, C4○D12, D42S3, S3×C10, C5×C4○D4, C23.8D6, S3×C2×C10, C5×C422C2, C5×C4○D12, C5×D42S3, C5×C23.8D6

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M 10N 10O 10P 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20AB 20AC ··· 20AJ 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 2 2 2 4 6 6 6 6 12 12 1 1 1 1 2 2 2 4 4 1 ··· 1 4 4 4 4 4 4 4 4 2 2 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 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D6 D6 C4○D4 C5×S3 C4○D12 S3×C10 S3×C10 C5×C4○D4 C5×C4○D12 D4⋊2S3 C5×D4⋊2S3 kernel C5×C23.8D6 Dic3×C20 C5×Dic3⋊C4 C5×C4⋊Dic3 C5×C6.D4 C15×C22⋊C4 C23.8D6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C6.D4 C3×C22⋊C4 C5×C22⋊C4 C2×C20 C22×C10 C30 C22⋊C4 C10 C2×C4 C23 C6 C2 C10 C2 # reps 1 1 2 1 2 1 4 4 8 4 8 4 1 2 1 6 4 4 8 4 24 16 2 8

Matrix representation of C5×C23.8D6 in GL4(𝔽61) generated by

 20 0 0 0 0 20 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 0 60 0 0 0 0 1 0 0 0 60 60
,
 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
,
 21 0 0 0 0 29 0 0 0 0 60 59 0 0 1 1
,
 0 29 0 0 21 0 0 0 0 0 11 22 0 0 50 50
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[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],[21,0,0,0,0,29,0,0,0,0,60,1,0,0,59,1],[0,21,0,0,29,0,0,0,0,0,11,50,0,0,22,50] >;

C5×C23.8D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._8D_6
% in TeX

G:=Group("C5xC2^3.8D6");
// GroupNames label

G:=SmallGroup(480,758);
// by ID

G=gap.SmallGroup(480,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,2606,891,15686]);
// 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=e^5>;
// generators/relations

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