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## G = C3×D4⋊Dic5order 480 = 25·3·5

### Direct product of C3 and D4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4⋊Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4⋊Dic5 — C3×D4⋊Dic5
 Lower central C5 — C10 — C20 — C3×D4⋊Dic5
 Upper central C1 — C2×C6 — C2×C12 — C6×D4

Generators and relations for C3×D4⋊Dic5
G = < a,b,c,d,e | a3=b4=c2=d10=1, e2=d5, 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: 272 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], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C24, C2×C12, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30 [×3], C30 [×2], D4⋊C4, C52C8, C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C3×C4⋊C4, C2×C24, C6×D4, C3×Dic5, C60 [×2], C2×C30, C2×C30 [×4], C2×C52C8, C4⋊Dic5, D4×C10, C3×D4⋊C4, C3×C52C8, C6×Dic5, C2×C60, D4×C15 [×2], D4×C15, C22×C30, D4⋊Dic5, C6×C52C8, C3×C4⋊Dic5, D4×C30, C3×D4⋊Dic5
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D8, SD16, Dic5 [×2], D10, C2×C12, C3×D4 [×2], C3×D5, D4⋊C4, C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C3×D8, C3×SD16, C3×Dic5 [×2], C6×D5, D4⋊D5, D4.D5, C23.D5, C3×D4⋊C4, C6×Dic5, C3×C5⋊D4 [×2], D4⋊Dic5, C3×D4⋊D5, C3×D4.D5, C3×C23.D5, C3×D4⋊Dic5

Smallest permutation representation of C3×D4⋊Dic5
On 240 points
Generators in S240
(1 95 55)(2 96 56)(3 97 57)(4 98 58)(5 99 59)(6 100 60)(7 91 51)(8 92 52)(9 93 53)(10 94 54)(11 102 62)(12 103 63)(13 104 64)(14 105 65)(15 106 66)(16 107 67)(17 108 68)(18 109 69)(19 110 70)(20 101 61)(21 208 168)(22 209 169)(23 210 170)(24 201 161)(25 202 162)(26 203 163)(27 204 164)(28 205 165)(29 206 166)(30 207 167)(31 111 71)(32 112 72)(33 113 73)(34 114 74)(35 115 75)(36 116 76)(37 117 77)(38 118 78)(39 119 79)(40 120 80)(41 124 81)(42 125 82)(43 126 83)(44 127 84)(45 128 85)(46 129 86)(47 130 87)(48 121 88)(49 122 89)(50 123 90)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)
(1 47 31 13)(2 48 32 14)(3 49 33 15)(4 50 34 16)(5 41 35 17)(6 42 36 18)(7 43 37 19)(8 44 38 20)(9 45 39 11)(10 46 40 12)(21 220 226 239)(22 211 227 240)(23 212 228 231)(24 213 229 232)(25 214 230 233)(26 215 221 234)(27 216 222 235)(28 217 223 236)(29 218 224 237)(30 219 225 238)(51 83 77 70)(52 84 78 61)(53 85 79 62)(54 86 80 63)(55 87 71 64)(56 88 72 65)(57 89 73 66)(58 90 74 67)(59 81 75 68)(60 82 76 69)(91 126 117 110)(92 127 118 101)(93 128 119 102)(94 129 120 103)(95 130 111 104)(96 121 112 105)(97 122 113 106)(98 123 114 107)(99 124 115 108)(100 125 116 109)(131 147 160 169)(132 148 151 170)(133 149 152 161)(134 150 153 162)(135 141 154 163)(136 142 155 164)(137 143 156 165)(138 144 157 166)(139 145 158 167)(140 146 159 168)(171 187 200 209)(172 188 191 210)(173 189 192 201)(174 190 193 202)(175 181 194 203)(176 182 195 204)(177 183 196 205)(178 184 197 206)(179 185 198 207)(180 186 199 208)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(91 110)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(100 109)(111 130)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)(131 160)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)(171 200)(172 191)(173 192)(174 193)(175 194)(176 195)(177 196)(178 197)(179 198)(180 199)(211 240)(212 231)(213 232)(214 233)(215 234)(216 235)(217 236)(218 237)(219 238)(220 239)
(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 6 140)(2 134 7 139)(3 133 8 138)(4 132 9 137)(5 131 10 136)(11 143 16 148)(12 142 17 147)(13 141 18 146)(14 150 19 145)(15 149 20 144)(21 130 26 125)(22 129 27 124)(23 128 28 123)(24 127 29 122)(25 126 30 121)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 179 56 174)(52 178 57 173)(53 177 58 172)(54 176 59 171)(55 175 60 180)(61 184 66 189)(62 183 67 188)(63 182 68 187)(64 181 69 186)(65 190 70 185)(71 194 76 199)(72 193 77 198)(73 192 78 197)(74 191 79 196)(75 200 80 195)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 219 96 214)(92 218 97 213)(93 217 98 212)(94 216 99 211)(95 215 100 220)(101 224 106 229)(102 223 107 228)(103 222 108 227)(104 221 109 226)(105 230 110 225)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 24A ··· 24H 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 24 ··· 24 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 1 1 2 2 20 20 2 2 1 ··· 1 4 4 4 4 10 10 10 10 2 ··· 2 4 ··· 4 2 2 2 2 20 20 20 20 2 2 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 ··· 4 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D5 D8 SD16 D10 Dic5 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C3×D8 C3×SD16 C6×D5 C3×Dic5 C3×C5⋊D4 C3×C5⋊D4 D4⋊D5 D4.D5 C3×D4⋊D5 C3×D4.D5 kernel C3×D4⋊Dic5 C6×C5⋊2C8 C3×C4⋊Dic5 D4×C30 D4⋊Dic5 D4×C15 C2×C5⋊2C8 C4⋊Dic5 D4×C10 C5×D4 C60 C2×C30 C6×D4 C30 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C10 C10 C2×C4 D4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 4 2 2 4 4 4 4 4 4 8 8 8 2 2 4 4

Matrix representation of C3×D4⋊Dic5 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 225 0 0 0 0 225
,
 1 0 0 0 0 1 0 0 0 0 240 215 0 0 204 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 37 240
,
 52 1 0 0 240 0 0 0 0 0 240 0 0 0 0 240
,
 181 240 0 0 227 60 0 0 0 0 0 12 0 0 20 0
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,240,204,0,0,215,1],[1,0,0,0,0,1,0,0,0,0,1,37,0,0,0,240],[52,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[181,227,0,0,240,60,0,0,0,0,0,20,0,0,12,0] >;

C3×D4⋊Dic5 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C3xD4:Dic5");
// GroupNames label

G:=SmallGroup(480,110);
// by ID

G=gap.SmallGroup(480,110);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=1,e^2=d^5,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