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G = C5×C23.16D6order 480 = 25·3·5

Direct product of C5 and C23.16D6

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

Aliases: C5×C23.16D6, Dic3⋊C47C10, (C4×Dic3)⋊9C10, (C2×Dic3)⋊3C20, (C2×C20).270D6, C22.6(S3×C20), C6.5(C22×C20), (Dic3×C20)⋊27C2, (C10×Dic3)⋊13C4, C23.16(S3×C10), Dic3.6(C2×C20), (C22×C10).86D6, C1526(C42⋊C2), C30.242(C4○D4), (C2×C30).397C23, C30.196(C22×C4), (C2×C60).414C22, C6.D4.1C10, C10.106(D42S3), (C22×Dic3).2C10, (C22×C30).112C22, (C10×Dic3).238C22, C2.7(S3×C2×C20), C10.132(S3×C2×C4), (C2×C6).4(C2×C20), C32(C5×C42⋊C2), C6.18(C5×C4○D4), (C2×C4).24(S3×C10), (C2×C10).65(C4×S3), C2.1(C5×D42S3), (C5×C22⋊C4).6S3, C22⋊C4.3(C5×S3), C22.12(S3×C2×C10), (C2×C12).52(C2×C10), (C2×C30).128(C2×C4), (C5×Dic3⋊C4)⋊29C2, (C15×C22⋊C4).9C2, (C3×C22⋊C4).3C10, (C22×C6).7(C2×C10), (Dic3×C2×C10).10C2, (C2×C6).18(C22×C10), (C5×Dic3).48(C2×C4), (C5×C6.D4).7C2, (C2×C10).331(C22×S3), (C2×Dic3).20(C2×C10), SmallGroup(480,756)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C23.16D6
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C2×C10 — C5×C23.16D6
C3C6 — C5×C23.16D6
C1C2×C10C5×C22⋊C4

Generators and relations for C5×C23.16D6
 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, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 276 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, C20 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×C20 [×2], C2×C20 [×8], C22×C10, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3 [×4], C5×Dic3 [×2], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×C20 [×2], C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4 [×2], C22×C20, C23.16D6, C10×Dic3 [×2], C10×Dic3 [×6], C2×C60 [×2], C22×C30, C5×C42⋊C2, Dic3×C20 [×2], C5×Dic3⋊C4 [×2], C5×C6.D4, C15×C22⋊C4, Dic3×C2×C10, C5×C23.16D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], C22×C4, C4○D4 [×2], C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C42⋊C2, C2×C20 [×6], C22×C10, S3×C2×C4, D42S3 [×2], S3×C10 [×3], C22×C20, C5×C4○D4 [×2], C23.16D6, S3×C20 [×2], S3×C2×C10, C5×C42⋊C2, S3×C2×C20, C5×D42S3 [×2], C5×C23.16D6

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

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

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

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

150 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I···4N5A5B5C5D6A6B6C6D6E10A···10L10M···10T12A12B12C12D15A15B15C15D20A···20P20Q···20AF20AG···20BD30A···30L30M···30T60A···60P
order1222223444444444···455556666610···1010···10121212121515151520···2020···2020···2030···3030···3060···60
size1111222222233336···61111222441···12···2444422222···23···36···62···24···44···4

150 irreducible representations

dim11111111111111222222222244
type+++++++++-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20S3D6D6C4○D4C4×S3C5×S3S3×C10S3×C10C5×C4○D4S3×C20D42S3C5×D42S3
kernelC5×C23.16D6Dic3×C20C5×Dic3⋊C4C5×C6.D4C15×C22⋊C4Dic3×C2×C10C10×Dic3C23.16D6C4×Dic3Dic3⋊C4C6.D4C3×C22⋊C4C22×Dic3C2×Dic3C5×C22⋊C4C2×C20C22×C10C30C2×C10C22⋊C4C2×C4C23C6C22C10C2
# reps12211184884443212144484161628

Matrix representation of C5×C23.16D6 in GL4(𝔽61) generated by

34000
03400
0090
0009
,
60000
06000
00600
00501
,
60000
06000
00600
00060
,
1000
0100
00600
00060
,
01100
505000
001159
00050
,
273100
43400
006039
0001
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,60,50,0,0,0,1],[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],[0,50,0,0,11,50,0,0,0,0,11,0,0,0,59,50],[27,4,0,0,31,34,0,0,0,0,60,0,0,0,39,1] >;

C5×C23.16D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{16}D_6
% in TeX

G:=Group("C5xC2^3.16D6");
// GroupNames label

G:=SmallGroup(480,756);
// by ID

G=gap.SmallGroup(480,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,891,226,15686]);
// 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,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations

׿
×
𝔽