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G = C5×C23.9D6order 480 = 25·3·5

Direct product of C5 and C23.9D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C23.9D6, D6⋊C45C10, D6.4(C5×D4), C4⋊Dic34C10, C6.19(D4×C10), (S3×C10).40D4, (C2×C20).272D6, C10.172(S3×D4), C30.355(C2×D4), C23.9(S3×C10), Dic3⋊C410C10, C6.D44C10, (C22×C10).21D6, C30.245(C4○D4), (C2×C60).327C22, (C2×C30).403C23, C10.115(C4○D12), C1529(C22.D4), C10.110(D42S3), (C22×C30).118C22, (C10×Dic3).139C22, C2.8(C5×S3×D4), (S3×C2×C4)⋊10C10, (S3×C2×C20)⋊26C2, C6.7(C5×C4○D4), (C5×D6⋊C4)⋊21C2, C22⋊C43(C5×S3), (C2×C4).6(S3×C10), (C5×C22⋊C4)⋊11S3, (C3×C22⋊C4)⋊5C10, (C2×C12).2(C2×C10), (C5×C4⋊Dic3)⋊22C2, C2.10(C5×C4○D12), C2.8(C5×D42S3), (C2×C3⋊D4).3C10, C22.42(S3×C2×C10), (C15×C22⋊C4)⋊19C2, C31(C5×C22.D4), (C5×Dic3⋊C4)⋊32C2, (C10×C3⋊D4).10C2, (C5×C6.D4)⋊20C2, (S3×C2×C10).108C22, (C2×C6).24(C22×C10), (C22×C6).13(C2×C10), (C22×S3).18(C2×C10), (C2×C10).337(C22×S3), (C2×Dic3).21(C2×C10), SmallGroup(480,762)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C23.9D6
C1C3C6C2×C6C2×C30S3×C2×C10S3×C2×C20 — C5×C23.9D6
C3C2×C6 — C5×C23.9D6
C1C2×C10C5×C22⋊C4

Generators and relations for C5×C23.9D6
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 372 in 156 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×2], C6 [×3], C6, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C10 [×3], C10 [×3], Dic3 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×3], C15, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C20 [×5], C2×C10, C2×C10 [×7], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C5×S3 [×2], C30 [×3], C30, C22.D4, C2×C20 [×2], C2×C20 [×5], C5×D4 [×2], C22×C10, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3 [×3], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×3], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, D4×C10, C23.9D6, S3×C20 [×2], C10×Dic3 [×3], C5×C3⋊D4 [×2], C2×C60 [×2], S3×C2×C10, C22×C30, C5×C22.D4, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4, C5×C6.D4, C15×C22⋊C4, S3×C2×C20, C10×C3⋊D4, C5×C23.9D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], C22×S3, C5×S3, C22.D4, C5×D4 [×2], C22×C10, C4○D12, S3×D4, D42S3, S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C23.9D6, S3×C2×C10, C5×C22.D4, C5×C4○D12, C5×S3×D4, C5×D42S3, C5×C23.9D6

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B5C5D6A6B6C6D6E10A···10L10M10N10O10P10Q···10X12A12B12C12D15A15B15C15D20A···20H20I20J20K20L20M···20T20U···20AB30A···30L30M···30T60A···60P
order12222223444444455556666610···101010101010···10121212121515151520···202020202020···2020···2030···3030···3060···60
size111146622246612121111222441···144446···6444422222···244446···612···122···24···44···4

120 irreducible representations

dim11111111111111112222222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D4D6D6C4○D4C5×S3C5×D4C4○D12S3×C10S3×C10C5×C4○D4C5×C4○D12S3×D4D42S3C5×S3×D4C5×D42S3
kernelC5×C23.9D6C5×Dic3⋊C4C5×C4⋊Dic3C5×D6⋊C4C5×C6.D4C15×C22⋊C4S3×C2×C20C10×C3⋊D4C23.9D6Dic3⋊C4C4⋊Dic3D6⋊C4C6.D4C3×C22⋊C4S3×C2×C4C2×C3⋊D4C5×C22⋊C4S3×C10C2×C20C22×C10C30C22⋊C4D6C10C2×C4C23C6C2C10C10C2C2
# reps1111111144444444122144848416161144

Matrix representation of C5×C23.9D6 in GL4(𝔽61) generated by

34000
03400
00580
00058
,
524300
18900
006059
0001
,
60000
06000
00600
00060
,
1000
0100
00600
00060
,
01100
505000
00500
001111
,
01100
11000
00500
00050
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,58,0,0,0,0,58],[52,18,0,0,43,9,0,0,0,0,60,0,0,0,59,1],[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],[0,50,0,0,11,50,0,0,0,0,50,11,0,0,0,11],[0,11,0,0,11,0,0,0,0,0,50,0,0,0,0,50] >;

C5×C23.9D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._9D_6
% in TeX

G:=Group("C5xC2^3.9D6");
// GroupNames label

G:=SmallGroup(480,762);
// by ID

G=gap.SmallGroup(480,762);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,288,926,891,15686]);
// Polycyclic

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

׿
×
𝔽