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G = C5×C23.14D6order 480 = 25·3·5

Direct product of C5 and C23.14D6

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

Aliases: C5×C23.14D6, (C6×D4)⋊9C10, (C2×C30)⋊21D4, D6⋊C415C10, (D4×C10)⋊12S3, (D4×C30)⋊25C2, Dic33(C5×D4), C6.51(D4×C10), C1539(C4⋊D4), (C5×Dic3)⋊18D4, C10.194(S3×D4), C30.434(C2×D4), (C2×C20).244D6, Dic3⋊C415C10, C23.14(S3×C10), (C22×C10).94D6, C30.256(C4○D4), C6.D412C10, (C2×C30).433C23, (C2×C60).429C22, (C22×Dic3)⋊6C10, C10.123(D42S3), (C22×C30).126C22, (C10×Dic3).230C22, (C2×C6)⋊3(C5×D4), C35(C5×C4⋊D4), C2.27(C5×S3×D4), (C2×D4)⋊5(C5×S3), (C5×D6⋊C4)⋊37C2, (C2×C3⋊D4)⋊6C10, C6.31(C5×C4○D4), C222(C5×C3⋊D4), (C10×C3⋊D4)⋊21C2, (C2×C4).17(S3×C10), (Dic3×C2×C10)⋊17C2, C2.15(C10×C3⋊D4), C22.61(S3×C2×C10), (C2×C10)⋊11(C3⋊D4), (C2×C12).63(C2×C10), (C5×Dic3⋊C4)⋊37C2, C2.18(C5×D42S3), C10.136(C2×C3⋊D4), (S3×C2×C10).72C22, (C5×C6.D4)⋊28C2, (C22×C6).21(C2×C10), (C2×C6).54(C22×C10), (C22×S3).11(C2×C10), (C2×C10).367(C22×S3), (C2×Dic3).38(C2×C10), SmallGroup(480,818)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C23.14D6
C1C3C6C2×C6C2×C30S3×C2×C10C10×C3⋊D4 — C5×C23.14D6
C3C2×C6 — C5×C23.14D6
C1C2×C10D4×C10

Generators and relations for C5×C23.14D6
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e6=1, f2=d, 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=de-1 >

Subgroups: 452 in 188 conjugacy classes, 70 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], C5, S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×2], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C5×S3, C30 [×3], C30 [×3], C4⋊D4, C2×C20, C2×C20 [×5], C5×D4 [×6], C22×C10 [×2], C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], C5×Dic3 [×2], C60, S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×5], C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, D4×C10, D4×C10 [×2], C23.14D6, C10×Dic3 [×3], C10×Dic3 [×2], C5×C3⋊D4 [×4], C2×C60, D4×C15 [×2], S3×C2×C10, C22×C30 [×2], C5×C4⋊D4, C5×Dic3⋊C4, C5×D6⋊C4, C5×C6.D4, Dic3×C2×C10, C10×C3⋊D4 [×2], D4×C30, C5×C23.14D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×4], C23, C10 [×7], D6 [×3], C2×D4 [×2], C4○D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C4⋊D4, C5×D4 [×4], C22×C10, S3×D4, D42S3, C2×C3⋊D4, S3×C10 [×3], D4×C10 [×2], C5×C4○D4, C23.14D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C4⋊D4, C5×S3×D4, C5×D42S3, C10×C3⋊D4, C5×C23.14D6

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B5C5D6A6B6C6D6E6F6G10A···10L10M···10T10U10V10W10X10Y10Z10AA10AB12A12B15A15B15C15D20A20B20C20D20E···20T20U20V20W20X30A···30L30M···30AB60A···60H
order1222222234444445555666666610···1010···1010101010101010101212151515152020202020···202020202030···3030···3060···60
size11112241224666612111122244441···12···244441212121244222244446···6121212122···24···44···4

120 irreducible representations

dim11111111111111222222222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10S3D4D4D6D6C4○D4C3⋊D4C5×S3C5×D4C5×D4S3×C10S3×C10C5×C4○D4C5×C3⋊D4S3×D4D42S3C5×S3×D4C5×D42S3
kernelC5×C23.14D6C5×Dic3⋊C4C5×D6⋊C4C5×C6.D4Dic3×C2×C10C10×C3⋊D4D4×C30C23.14D6Dic3⋊C4D6⋊C4C6.D4C22×Dic3C2×C3⋊D4C6×D4D4×C10C5×Dic3C2×C30C2×C20C22×C10C30C2×C10C2×D4Dic3C2×C6C2×C4C23C6C22C10C10C2C2
# reps111112144444841221224488488161144

Matrix representation of C5×C23.14D6 in GL6(𝔽61)

3400000
0340000
0058000
0005800
000090
000009
,
6000000
0600000
0014000
0006000
000001
000010
,
100000
010000
0060000
0006000
000010
000001
,
100000
010000
0060000
0006000
0000600
0000060
,
6010000
6000000
001000
0036000
000010
0000060
,
48220000
9130000
00111300
0005000
000001
0000600

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,40,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[48,9,0,0,0,0,22,13,0,0,0,0,0,0,11,0,0,0,0,0,13,50,0,0,0,0,0,0,0,60,0,0,0,0,1,0] >;

C5×C23.14D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{14}D_6
% in TeX

G:=Group("C5xC2^3.14D6");
// GroupNames label

G:=SmallGroup(480,818);
// by ID

G=gap.SmallGroup(480,818);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,891,15686]);
// Polycyclic

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

׿
×
𝔽