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G = C3×D205C4order 480 = 25·3·5

Direct product of C3 and D205C4

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

Aliases: C3×D205C4, D205C12, C30.31D8, C6.17D40, C60.216D4, C30.21SD16, (C2×C40)⋊2C6, (C2×C24)⋊2D5, (C2×C120)⋊2C2, C4⋊Dic51C6, C10.5(C3×D8), C4.8(D5×C12), C2.2(C3×D40), (C3×D20)⋊14C4, (C2×D20).1C6, C20.45(C3×D4), C12.65(C4×D5), (C2×C6).51D20, C20.39(C2×C12), C60.200(C2×C4), (C6×D20).10C2, (C2×C30).111D4, C10.3(C3×SD16), C1515(D4⋊C4), (C2×C12).425D10, C6.11(C40⋊C2), C22.10(C3×D20), C12.113(C5⋊D4), C30.86(C22⋊C4), (C2×C60).505C22, C6.39(D10⋊C4), (C2×C8)⋊2(C3×D5), C53(C3×D4⋊C4), (C2×C4).72(C6×D5), C2.3(C3×C40⋊C2), C4.20(C3×C5⋊D4), (C2×C20).88(C2×C6), (C3×C4⋊Dic5)⋊13C2, (C2×C10).15(C3×D4), C2.8(C3×D10⋊C4), C10.18(C3×C22⋊C4), SmallGroup(480,99)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D205C4
C1C5C10C2×C10C2×C20C2×C60C6×D20 — C3×D205C4
C5C10C20 — C3×D205C4
C1C2×C6C2×C12C2×C24

Generators and relations for C3×D205C4
 G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b3c >

Subgroups: 464 in 100 conjugacy classes, 46 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 [×3], C23, D5 [×2], C10 [×3], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], D10 [×4], C2×C10, C24, C2×C12, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], C30 [×3], D4⋊C4, C40, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C3×Dic5, C60 [×2], C6×D5 [×4], C2×C30, C4⋊Dic5, C2×C40, C2×D20, C3×D4⋊C4, C120, C3×D20 [×2], C3×D20, C6×Dic5, C2×C60, D5×C2×C6, D205C4, C3×C4⋊Dic5, C2×C120, C6×D20, C3×D205C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D8, SD16, D10, C2×C12, C3×D4 [×2], C3×D5, D4⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×D5, C40⋊C2, D40, D10⋊C4, C3×D4⋊C4, D5×C12, C3×D20, C3×C5⋊D4, D205C4, C3×C40⋊C2, C3×D40, C3×D10⋊C4, C3×D205C4

Smallest permutation representation of C3×D205C4
On 240 points
Generators in S240
(1 188 40)(2 189 21)(3 190 22)(4 191 23)(5 192 24)(6 193 25)(7 194 26)(8 195 27)(9 196 28)(10 197 29)(11 198 30)(12 199 31)(13 200 32)(14 181 33)(15 182 34)(16 183 35)(17 184 36)(18 185 37)(19 186 38)(20 187 39)(41 79 141)(42 80 142)(43 61 143)(44 62 144)(45 63 145)(46 64 146)(47 65 147)(48 66 148)(49 67 149)(50 68 150)(51 69 151)(52 70 152)(53 71 153)(54 72 154)(55 73 155)(56 74 156)(57 75 157)(58 76 158)(59 77 159)(60 78 160)(81 173 107)(82 174 108)(83 175 109)(84 176 110)(85 177 111)(86 178 112)(87 179 113)(88 180 114)(89 161 115)(90 162 116)(91 163 117)(92 164 118)(93 165 119)(94 166 120)(95 167 101)(96 168 102)(97 169 103)(98 170 104)(99 171 105)(100 172 106)(121 235 209)(122 236 210)(123 237 211)(124 238 212)(125 239 213)(126 240 214)(127 221 215)(128 222 216)(129 223 217)(130 224 218)(131 225 219)(132 226 220)(133 227 201)(134 228 202)(135 229 203)(136 230 204)(137 231 205)(138 232 206)(139 233 207)(140 234 208)
(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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(81 132)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 206)(102 205)(103 204)(104 203)(105 202)(106 201)(107 220)(108 219)(109 218)(110 217)(111 216)(112 215)(113 214)(114 213)(115 212)(116 211)(117 210)(118 209)(119 208)(120 207)(141 200)(142 199)(143 198)(144 197)(145 196)(146 195)(147 194)(148 193)(149 192)(150 191)(151 190)(152 189)(153 188)(154 187)(155 186)(156 185)(157 184)(158 183)(159 182)(160 181)(161 238)(162 237)(163 236)(164 235)(165 234)(166 233)(167 232)(168 231)(169 230)(170 229)(171 228)(172 227)(173 226)(174 225)(175 224)(176 223)(177 222)(178 221)(179 240)(180 239)
(1 121 62 98)(2 140 63 97)(3 139 64 96)(4 138 65 95)(5 137 66 94)(6 136 67 93)(7 135 68 92)(8 134 69 91)(9 133 70 90)(10 132 71 89)(11 131 72 88)(12 130 73 87)(13 129 74 86)(14 128 75 85)(15 127 76 84)(16 126 77 83)(17 125 78 82)(18 124 79 81)(19 123 80 100)(20 122 61 99)(21 208 45 103)(22 207 46 102)(23 206 47 101)(24 205 48 120)(25 204 49 119)(26 203 50 118)(27 202 51 117)(28 201 52 116)(29 220 53 115)(30 219 54 114)(31 218 55 113)(32 217 56 112)(33 216 57 111)(34 215 58 110)(35 214 59 109)(36 213 60 108)(37 212 41 107)(38 211 42 106)(39 210 43 105)(40 209 44 104)(141 173 185 238)(142 172 186 237)(143 171 187 236)(144 170 188 235)(145 169 189 234)(146 168 190 233)(147 167 191 232)(148 166 192 231)(149 165 193 230)(150 164 194 229)(151 163 195 228)(152 162 196 227)(153 161 197 226)(154 180 198 225)(155 179 199 224)(156 178 200 223)(157 177 181 222)(158 176 182 221)(159 175 183 240)(160 174 184 239)

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

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

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

138 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D5A5B6A···6F6G6H6I6J8A8B8C8D10A···10F12A12B12C12D12E12F12G12H15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order122222334444556···66666888810···1012121212121212121515151520···2024···2430···3040···4060···60120···120
size1111202011222020221···12020202022222···222222020202022222···22···22···22···22···22···2

138 irreducible representations

dim11111111112222222222222222222222
type+++++++++++
imageC1C2C2C2C3C4C6C6C6C12D4D4D5D8SD16D10C3×D4C3×D4C3×D5C4×D5C5⋊D4D20C3×D8C3×SD16C6×D5C40⋊C2D40D5×C12C3×C5⋊D4C3×D20C3×C40⋊C2C3×D40
kernelC3×D205C4C3×C4⋊Dic5C2×C120C6×D20D205C4C3×D20C4⋊Dic5C2×C40C2×D20D20C60C2×C30C2×C24C30C30C2×C12C20C2×C10C2×C8C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps1111242228112222224444444888881616

Matrix representation of C3×D205C4 in GL4(𝔽241) generated by

225000
022500
00150
00015
,
240000
024000
004144
0078119
,
1000
4024000
0015678
0020485
,
23213300
32900
00116210
00162125
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,15,0,0,0,0,15],[240,0,0,0,0,240,0,0,0,0,41,78,0,0,44,119],[1,40,0,0,0,240,0,0,0,0,156,204,0,0,78,85],[232,32,0,0,133,9,0,0,0,0,116,162,0,0,210,125] >;

C3×D205C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_5C_4
% in TeX

G:=Group("C3xD20:5C4");
// GroupNames label

G:=SmallGroup(480,99);
// by ID

G=gap.SmallGroup(480,99);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,260,1683,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^3*c>;
// generators/relations

׿
×
𝔽