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G = C5×C23.21D6order 480 = 25·3·5

Direct product of C5 and C23.21D6

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

Aliases: C5×C23.21D6, D6⋊C46C10, C6.6(D4×C10), C4⋊Dic35C10, C2.8(C10×D12), (C2×C30).91D4, C10.77(C2×D12), C30.293(C2×D4), (C2×C20).235D6, (C2×C10).26D12, C22.4(C5×D12), C23.21(S3×C10), (C22×C10).91D6, C30.247(C4○D4), (C2×C60).330C22, (C2×C30).406C23, (C22×Dic3)⋊2C10, C1530(C22.D4), C10.112(D42S3), (C22×C30).121C22, (C10×Dic3).141C22, (C2×C6).4(C5×D4), (C5×D6⋊C4)⋊22C2, C22⋊C46(C5×S3), (C2×C4).8(S3×C10), C6.22(C5×C4○D4), (C3×C22⋊C4)⋊4C10, (C5×C22⋊C4)⋊14S3, (C2×C12).4(C2×C10), (C5×C4⋊Dic3)⋊23C2, (Dic3×C2×C10)⋊13C2, (C2×C3⋊D4).5C10, C22.45(S3×C2×C10), (C15×C22⋊C4)⋊18C2, C32(C5×C22.D4), C2.10(C5×D42S3), (C10×C3⋊D4).12C2, (S3×C2×C10).67C22, (C22×S3).6(C2×C10), (C2×C6).27(C22×C10), (C22×C6).16(C2×C10), (C2×Dic3).8(C2×C10), (C2×C10).340(C22×S3), SmallGroup(480,765)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C23.21D6
C1C3C6C2×C6C2×C30S3×C2×C10C10×C3⋊D4 — C5×C23.21D6
C3C2×C6 — C5×C23.21D6
C1C2×C10C5×C22⋊C4

Generators and relations for C5×C23.21D6
 G = < a,b,c,d,e | a5=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 372 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C10, C10 [×2], C10 [×3], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C5×S3, C30, C30 [×2], C30 [×2], C22.D4, C2×C20 [×2], C2×C20 [×5], C5×D4 [×2], C22×C10, C22×C10, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×3], C60 [×2], S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, D4×C10, C23.21D6, C10×Dic3, C10×Dic3 [×2], C10×Dic3 [×2], C5×C3⋊D4 [×2], C2×C60 [×2], S3×C2×C10, C22×C30, C5×C22.D4, C5×C4⋊Dic3 [×2], C5×D6⋊C4 [×2], C15×C22⋊C4, Dic3×C2×C10, C10×C3⋊D4, C5×C23.21D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C22.D4, C5×D4 [×2], C22×C10, C2×D12, D42S3 [×2], S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C23.21D6, C5×D12 [×2], S3×C2×C10, C5×C22.D4, C10×D12, C5×D42S3 [×2], C5×C23.21D6

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B5C5D6A6B6C6D6E10A···10L10M···10T10U10V10W10X12A12B12C12D15A15B15C15D20A···20H20I···20X20Y20Z20AA20AB30A···30L30M···30T60A···60P
order12222223444444455556666610···1010···1010101010121212121515151520···2020···202020202030···3030···3060···60
size111122122446666121111222441···12···212121212444422224···46···6121212122···24···44···4

120 irreducible representations

dim11111111111122222222222244
type+++++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D6D6C4○D4D12C5×S3C5×D4S3×C10S3×C10C5×C4○D4C5×D12D42S3C5×D42S3
kernelC5×C23.21D6C5×C4⋊Dic3C5×D6⋊C4C15×C22⋊C4Dic3×C2×C10C10×C3⋊D4C23.21D6C4⋊Dic3D6⋊C4C3×C22⋊C4C22×Dic3C2×C3⋊D4C5×C22⋊C4C2×C30C2×C20C22×C10C30C2×C10C22⋊C4C2×C6C2×C4C23C6C22C10C2
# reps1221114884441221444884161628

Matrix representation of C5×C23.21D6 in GL6(𝔽61)

3400000
0340000
0020000
0002000
000090
000009
,
100000
010000
001000
0006000
000010
000001
,
100000
010000
0060000
0006000
000010
000001
,
010000
6000000
000100
0060000
0000160
000010
,
3420000
2270000
0011000
0001100
00003838
00001523

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[34,2,0,0,0,0,2,27,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,38,15,0,0,0,0,38,23] >;

C5×C23.21D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{21}D_6
% in TeX

G:=Group("C5xC2^3.21D6");
// GroupNames label

G:=SmallGroup(480,765);
// by ID

G=gap.SmallGroup(480,765);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,891,646,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

׿
×
𝔽