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G = C5×D4×Dic3order 480 = 25·3·5

Direct product of C5, D4 and Dic3

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

Aliases: C5×D4×Dic3, C35(D4×C20), C1545(C4×D4), C123(C2×C20), C6031(C2×C4), (C3×D4)⋊3C20, (D4×C15)⋊15C4, (C6×D4).4C10, C209(C2×Dic3), C41(C10×Dic3), C6.37(D4×C10), C4⋊Dic313C10, (C4×Dic3)⋊4C10, (D4×C10).14S3, (D4×C30).14C2, C10.191(S3×D4), C30.373(C2×D4), (C2×C20).360D6, (Dic3×C20)⋊16C2, C6.D47C10, C23.22(S3×C10), C6.25(C22×C20), C222(C10×Dic3), (C22×C10).92D6, C30.252(C4○D4), (C2×C30).428C23, (C2×C60).360C22, C30.232(C22×C4), (C22×Dic3)⋊4C10, C10.119(D42S3), C10.48(C22×Dic3), (C22×C30).122C22, (C10×Dic3).227C22, C2.5(C5×S3×D4), (C2×C6)⋊3(C2×C20), (C2×C30)⋊31(C2×C4), (C2×D4).7(C5×S3), C6.27(C5×C4○D4), C2.6(Dic3×C2×C10), (C2×C4).49(S3×C10), (C5×C4⋊Dic3)⋊31C2, (Dic3×C2×C10)⋊15C2, C2.5(C5×D42S3), C22.25(S3×C2×C10), (C2×C12).33(C2×C10), (C2×C10)⋊11(C2×Dic3), (C5×C6.D4)⋊23C2, (C2×C6).49(C22×C10), (C22×C6).17(C2×C10), (C2×C10).362(C22×S3), (C2×Dic3).35(C2×C10), SmallGroup(480,813)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D4×Dic3
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C2×C10 — C5×D4×Dic3
C3C6 — C5×D4×Dic3
C1C2×C10D4×C10

Generators and relations for C5×D4×Dic3
 G = < a,b,c,d,e | a5=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 356 in 188 conjugacy classes, 102 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C20 [×2], C20 [×5], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×4], C22×C6 [×2], C30 [×3], C30 [×4], C4×D4, C2×C20, C2×C20 [×8], C5×D4 [×4], C22×C10 [×2], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, C5×Dic3 [×2], C5×Dic3 [×3], C60 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, D4×Dic3, C10×Dic3 [×2], C10×Dic3 [×2], C10×Dic3 [×4], C2×C60, D4×C15 [×4], C22×C30 [×2], D4×C20, Dic3×C20, C5×C4⋊Dic3, C5×C6.D4 [×2], Dic3×C2×C10 [×2], D4×C30, C5×D4×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], C23, C10 [×7], Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×S3, C5×S3, C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, S3×D4, D42S3, C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C22×C20, D4×C10, C5×C4○D4, D4×Dic3, C10×Dic3 [×6], S3×C2×C10, D4×C20, C5×S3×D4, C5×D42S3, Dic3×C2×C10, C5×D4×Dic3

Smallest permutation representation of C5×D4×Dic3
On 240 points
Generators in S240
(1 60 48 36 24)(2 55 43 31 19)(3 56 44 32 20)(4 57 45 33 21)(5 58 46 34 22)(6 59 47 35 23)(7 230 218 206 194)(8 231 219 207 195)(9 232 220 208 196)(10 233 221 209 197)(11 234 222 210 198)(12 229 217 205 193)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 126 109 97 85)(74 121 110 98 86)(75 122 111 99 87)(76 123 112 100 88)(77 124 113 101 89)(78 125 114 102 90)(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)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(1 69 16 73)(2 70 17 74)(3 71 18 75)(4 72 13 76)(5 67 14 77)(6 68 15 78)(7 180 237 185)(8 175 238 186)(9 176 239 181)(10 177 240 182)(11 178 235 183)(12 179 236 184)(19 82 29 86)(20 83 30 87)(21 84 25 88)(22 79 26 89)(23 80 27 90)(24 81 28 85)(31 94 41 98)(32 95 42 99)(33 96 37 100)(34 91 38 101)(35 92 39 102)(36 93 40 97)(43 106 53 110)(44 107 54 111)(45 108 49 112)(46 103 50 113)(47 104 51 114)(48 105 52 109)(55 118 65 121)(56 119 66 122)(57 120 61 123)(58 115 62 124)(59 116 63 125)(60 117 64 126)(127 190 138 195)(128 191 133 196)(129 192 134 197)(130 187 135 198)(131 188 136 193)(132 189 137 194)(139 202 150 207)(140 203 145 208)(141 204 146 209)(142 199 147 210)(143 200 148 205)(144 201 149 206)(151 214 162 219)(152 215 157 220)(153 216 158 221)(154 211 159 222)(155 212 160 217)(156 213 161 218)(163 226 174 231)(164 227 169 232)(165 228 170 233)(166 223 171 234)(167 224 172 229)(168 225 173 230)
(1 4)(2 5)(3 6)(7 240)(8 235)(9 236)(10 237)(11 238)(12 239)(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 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(67 74)(68 75)(69 76)(70 77)(71 78)(72 73)(79 86)(80 87)(81 88)(82 89)(83 90)(84 85)(91 98)(92 99)(93 100)(94 101)(95 102)(96 97)(103 110)(104 111)(105 112)(106 113)(107 114)(108 109)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(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)(187 195)(188 196)(189 197)(190 198)(191 193)(192 194)(199 207)(200 208)(201 209)(202 210)(203 205)(204 206)(211 219)(212 220)(213 221)(214 222)(215 217)(216 218)(223 231)(224 232)(225 233)(226 234)(227 229)(228 230)
(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)
(1 129 4 132)(2 128 5 131)(3 127 6 130)(7 126 10 123)(8 125 11 122)(9 124 12 121)(13 137 16 134)(14 136 17 133)(15 135 18 138)(19 140 22 143)(20 139 23 142)(21 144 24 141)(25 149 28 146)(26 148 29 145)(27 147 30 150)(31 152 34 155)(32 151 35 154)(33 156 36 153)(37 161 40 158)(38 160 41 157)(39 159 42 162)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 173 52 170)(50 172 53 169)(51 171 54 174)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 185 64 182)(62 184 65 181)(63 183 66 186)(67 188 70 191)(68 187 71 190)(69 192 72 189)(73 197 76 194)(74 196 77 193)(75 195 78 198)(79 200 82 203)(80 199 83 202)(81 204 84 201)(85 209 88 206)(86 208 89 205)(87 207 90 210)(91 212 94 215)(92 211 95 214)(93 216 96 213)(97 221 100 218)(98 220 101 217)(99 219 102 222)(103 224 106 227)(104 223 107 226)(105 228 108 225)(109 233 112 230)(110 232 113 229)(111 231 114 234)(115 236 118 239)(116 235 119 238)(117 240 120 237)

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

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

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

150 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G···4L5A5B5C5D6A6B6C6D6E6F6G10A···10L10M···10AB12A12B15A15B15C15D20A···20H20I···20X20Y···20AV30A···30L30M···30AB60A···60H
order1222222234444444···45555666666610···1010···1012121515151520···2020···2020···2030···3030···3060···60
size1111222222233336···6111122244441···12···24422222···23···36···62···24···44···4

150 irreducible representations

dim111111111111112222222222224444
type+++++++++-++-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20S3D4D6Dic3D6C4○D4C5×S3C5×D4S3×C10C5×Dic3S3×C10C5×C4○D4S3×D4D42S3C5×S3×D4C5×D42S3
kernelC5×D4×Dic3Dic3×C20C5×C4⋊Dic3C5×C6.D4Dic3×C2×C10D4×C30D4×C15D4×Dic3C4×Dic3C4⋊Dic3C6.D4C22×Dic3C6×D4C3×D4D4×C10C5×Dic3C2×C20C5×D4C22×C10C30C2×D4Dic3C2×C4D4C23C6C10C10C2C2
# reps11122184448843212142248416881144

Matrix representation of C5×D4×Dic3 in GL5(𝔽61)

200000
09000
00900
000580
000058
,
600000
006000
01000
000600
000060
,
10000
01000
006000
00010
00001
,
600000
01000
00100
000601
000600
,
500000
01000
00100
0002544
000836

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,58,0,0,0,0,0,58],[60,0,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,1,0],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,25,8,0,0,0,44,36] >;

C5×D4×Dic3 in GAP, Magma, Sage, TeX

C_5\times D_4\times {\rm Dic}_3
% in TeX

G:=Group("C5xD4xDic3");
// GroupNames label

G:=SmallGroup(480,813);
// by ID

G=gap.SmallGroup(480,813);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,891,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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