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G = C5×D4⋊Dic3order 480 = 25·3·5

Direct product of C5 and D4⋊Dic3

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

Aliases: C5×D4⋊Dic3, C30.57D8, C60.140D4, C30.32SD16, (C3×D4)⋊1C20, C6.12(C5×D8), C12.7(C5×D4), (D4×C15)⋊13C4, C12.7(C2×C20), D41(C5×Dic3), (C5×D4)⋊7Dic3, (D4×C10).6S3, (C6×D4).1C10, C6.5(C5×SD16), C60.177(C2×C4), C4⋊Dic310C10, (D4×C30).11C2, (C2×C20).350D6, (C2×C30).176D4, C4.1(C10×Dic3), C1519(D4⋊C4), C10.28(D4⋊S3), C20.91(C3⋊D4), C20.51(C2×Dic3), C10.12(D4.S3), (C2×C60).345C22, C30.117(C22⋊C4), C10.33(C6.D4), (C2×C3⋊C8)⋊2C10, (C10×C3⋊C8)⋊16C2, C33(C5×D4⋊C4), C2.3(C5×D4⋊S3), (C2×D4).1(C5×S3), (C2×C6).33(C5×D4), C4.12(C5×C3⋊D4), C2.3(C5×D4.S3), (C2×C4).38(S3×C10), (C5×C4⋊Dic3)⋊28C2, C6.13(C5×C22⋊C4), (C2×C12).15(C2×C10), C2.3(C5×C6.D4), C22.17(C5×C3⋊D4), (C2×C10).89(C3⋊D4), SmallGroup(480,151)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D4⋊Dic3
C1C3C6C12C2×C12C2×C60C5×C4⋊Dic3 — C5×D4⋊Dic3
C3C6C12 — C5×D4⋊Dic3
C1C2×C10C2×C20D4×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=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 228 in 100 conjugacy classes, 50 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C10 [×3], C10 [×2], Dic3, C12 [×2], C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, C2×C10, C2×C10 [×4], C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30 [×3], C30 [×2], D4⋊C4, C40, C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C3⋊C8, C4⋊Dic3, C6×D4, C5×Dic3, C60 [×2], C2×C30, C2×C30 [×4], C5×C4⋊C4, C2×C40, D4×C10, D4⋊Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C5×D4⋊C4, C10×C3⋊C8, C5×C4⋊Dic3, D4×C30, C5×D4⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], Dic3 [×2], D6, C22⋊C4, D8, SD16, C20 [×2], C2×C10, C2×Dic3, C3⋊D4 [×2], C5×S3, D4⋊C4, C2×C20, C5×D4 [×2], D4⋊S3, D4.S3, C6.D4, C5×Dic3 [×2], S3×C10, C5×C22⋊C4, C5×D8, C5×SD16, D4⋊Dic3, C10×Dic3, C5×C3⋊D4 [×2], C5×D4⋊C4, C5×D4⋊S3, C5×D4.S3, C5×C6.D4, C5×D4⋊Dic3

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B5C5D6A6B6C6D6E6F6G8A8B8C8D10A···10L10M···10T12A12B15A15B15C15D20A···20H20I···20P30A···30L30M···30AB40A···40P60A···60H
order1222223444455556666666888810···1010···1012121515151520···2020···2030···3030···3040···4060···60
size11114422212121111222444466661···14···44422222···212···122···24···46···64···4

120 irreducible representations

dim11111111112222222222222222224444
type++++++++-++-
imageC1C2C2C2C4C5C10C10C10C20S3D4D4D6Dic3D8SD16C3⋊D4C3⋊D4C5×S3C5×D4C5×D4S3×C10C5×Dic3C5×D8C5×SD16C5×C3⋊D4C5×C3⋊D4D4⋊S3D4.S3C5×D4⋊S3C5×D4.S3
kernelC5×D4⋊Dic3C10×C3⋊C8C5×C4⋊Dic3D4×C30D4×C15D4⋊Dic3C2×C3⋊C8C4⋊Dic3C6×D4C3×D4D4×C10C60C2×C30C2×C20C5×D4C30C30C20C2×C10C2×D4C12C2×C6C2×C4D4C6C6C4C22C10C10C2C2
# reps111144444161111222224444888881144

Matrix representation of C5×D4⋊Dic3 in GL5(𝔽241)

10000
098000
009800
000870
000087
,
10000
0240000
0024000
0001239
0001240
,
2400000
0240000
00100
0001239
0000240
,
2400000
0225000
001500
00010
00001
,
1770000
00100
01000
0000219
0002300

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,98,0,0,0,0,0,98,0,0,0,0,0,87,0,0,0,0,0,87],[1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,1,1,0,0,0,239,240],[240,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,239,240],[240,0,0,0,0,0,225,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[177,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,230,0,0,0,219,0] >;

C5×D4⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C5xD4:Dic3");
// GroupNames label

G:=SmallGroup(480,151);
// by ID

G=gap.SmallGroup(480,151);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,4204,2111,102,15686]);
// 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=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽