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G = C5×C23.8D6order 480 = 25·3·5

Direct product of C5 and C23.8D6

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

Aliases: C5×C23.8D6, C4⋊Dic33C10, Dic3⋊C48C10, (C2×C20).233D6, C23.8(S3×C10), (Dic3×C20)⋊28C2, (C4×Dic3)⋊10C10, (C22×C10).20D6, C1520(C422C2), C30.203(C4○D4), (C2×C60).326C22, (C2×C30).399C23, C6.D4.3C10, C10.114(C4○D12), C10.108(D42S3), (C22×C30).114C22, (C10×Dic3).138C22, C32(C5×C422C2), C6.20(C5×C4○D4), C2.9(C5×C4○D12), (C2×C4).25(S3×C10), (C5×C4⋊Dic3)⋊21C2, C2.7(C5×D42S3), C22⋊C4.2(C5×S3), (C5×C22⋊C4).5S3, C22.40(S3×C2×C10), (C2×C12).53(C2×C10), (C5×Dic3⋊C4)⋊30C2, (C15×C22⋊C4).7C2, (C3×C22⋊C4).2C10, (C22×C6).9(C2×C10), (C2×C6).20(C22×C10), (C2×Dic3).6(C2×C10), (C5×C6.D4).9C2, (C2×C10).333(C22×S3), SmallGroup(480,758)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C23.8D6
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C20 — C5×C23.8D6
C3C2×C6 — C5×C23.8D6
C1C2×C10C5×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 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C20 [×6], C2×C10, C2×C10 [×3], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C30 [×3], C30, C422C2, C2×C20 [×2], C2×C20 [×4], C22×C10, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C5×Dic3 [×4], C60 [×2], C2×C30, C2×C30 [×3], C4×C20, C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C23.8D6, C10×Dic3 [×4], C2×C60 [×2], C22×C30, C5×C422C2, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×C6.D4 [×2], C15×C22⋊C4, C5×C23.8D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4 [×3], C2×C10 [×7], C22×S3, C5×S3, C422C2, C22×C10, C4○D12, D42S3 [×2], S3×C10 [×3], C5×C4○D4 [×3], C23.8D6, S3×C2×C10, C5×C422C2, C5×C4○D12, C5×D42S3 [×2], C5×C23.8D6

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B5C5D6A6B6C6D6E10A···10L10M10N10O10P12A12B12C12D15A15B15C15D20A···20H20I20J20K20L20M···20AB20AC···20AJ30A···30L30M···30T60A···60P
order12222344444444455556666610···1010101010121212121515151520···202020202020···2020···2030···3030···3060···60
size111142224666612121111222441···14444444422222···244446···612···122···24···44···4

120 irreducible representations

dim111111111111222222222244
type+++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D6D6C4○D4C5×S3C4○D12S3×C10S3×C10C5×C4○D4C5×C4○D12D42S3C5×D42S3
kernelC5×C23.8D6Dic3×C20C5×Dic3⋊C4C5×C4⋊Dic3C5×C6.D4C15×C22⋊C4C23.8D6C4×Dic3Dic3⋊C4C4⋊Dic3C6.D4C3×C22⋊C4C5×C22⋊C4C2×C20C22×C10C30C22⋊C4C10C2×C4C23C6C2C10C2
# reps11212144848412164484241628

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

20000
02000
00340
00034
,
1000
06000
0010
006060
,
60000
06000
00600
00060
,
1000
0100
00600
00060
,
21000
02900
006059
0011
,
02900
21000
001122
005050
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

׿
×
𝔽