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## G = C3×D20.3C4order 480 = 25·3·5

### Direct product of C3 and D20.3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D20.3C4
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×C4○D20 — C3×D20.3C4
 Lower central C5 — C10 — C3×D20.3C4
 Upper central C1 — C24 — C2×C24

Generators and relations for C3×D20.3C4
G = < a,b,c,d | a3=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 320 in 124 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, C6, C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C15, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C24 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C3×D5 [×2], C30, C30, C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, C2×C24, C2×C24 [×2], C3×M4(2) [×3], C3×C4○D4, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, C3×C8○D4, C3×C52C8 [×2], C120 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C2×C60, D20.3C4, D5×C24 [×2], C3×C8⋊D5 [×2], C3×C4.Dic5, C2×C120, C3×C4○D20, C3×D20.3C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, D10 [×3], C2×C12 [×6], C22×C6, C3×D5, C8○D4, C4×D5 [×2], C22×D5, C22×C12, C6×D5 [×3], C2×C4×D5, C3×C8○D4, D5×C12 [×2], D5×C2×C6, D20.3C4, D5×C2×C12, C3×D20.3C4

Smallest permutation representation of C3×D20.3C4
On 240 points
Generators in S240
(1 154 26)(2 155 27)(3 156 28)(4 157 29)(5 158 30)(6 159 31)(7 160 32)(8 141 33)(9 142 34)(10 143 35)(11 144 36)(12 145 37)(13 146 38)(14 147 39)(15 148 40)(16 149 21)(17 150 22)(18 151 23)(19 152 24)(20 153 25)(41 196 109)(42 197 110)(43 198 111)(44 199 112)(45 200 113)(46 181 114)(47 182 115)(48 183 116)(49 184 117)(50 185 118)(51 186 119)(52 187 120)(53 188 101)(54 189 102)(55 190 103)(56 191 104)(57 192 105)(58 193 106)(59 194 107)(60 195 108)(61 84 228)(62 85 229)(63 86 230)(64 87 231)(65 88 232)(66 89 233)(67 90 234)(68 91 235)(69 92 236)(70 93 237)(71 94 238)(72 95 239)(73 96 240)(74 97 221)(75 98 222)(76 99 223)(77 100 224)(78 81 225)(79 82 226)(80 83 227)(121 211 173)(122 212 174)(123 213 175)(124 214 176)(125 215 177)(126 216 178)(127 217 179)(128 218 180)(129 219 161)(130 220 162)(131 201 163)(132 202 164)(133 203 165)(134 204 166)(135 205 167)(136 206 168)(137 207 169)(138 208 170)(139 209 171)(140 210 172)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 140)(141 146)(142 145)(143 144)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 200)(196 199)(197 198)(201 218)(202 217)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)(219 220)(221 238)(222 237)(223 236)(224 235)(225 234)(226 233)(227 232)(228 231)(229 230)(239 240)
(1 63 220 43 11 73 210 53)(2 64 201 44 12 74 211 54)(3 65 202 45 13 75 212 55)(4 66 203 46 14 76 213 56)(5 67 204 47 15 77 214 57)(6 68 205 48 16 78 215 58)(7 69 206 49 17 79 216 59)(8 70 207 50 18 80 217 60)(9 71 208 51 19 61 218 41)(10 72 209 52 20 62 219 42)(21 225 125 106 31 235 135 116)(22 226 126 107 32 236 136 117)(23 227 127 108 33 237 137 118)(24 228 128 109 34 238 138 119)(25 229 129 110 35 239 139 120)(26 230 130 111 36 240 140 101)(27 231 131 112 37 221 121 102)(28 232 132 113 38 222 122 103)(29 233 133 114 39 223 123 104)(30 234 134 115 40 224 124 105)(81 177 193 159 91 167 183 149)(82 178 194 160 92 168 184 150)(83 179 195 141 93 169 185 151)(84 180 196 142 94 170 186 152)(85 161 197 143 95 171 187 153)(86 162 198 144 96 172 188 154)(87 163 199 145 97 173 189 155)(88 164 200 146 98 174 190 156)(89 165 181 147 99 175 191 157)(90 166 182 148 100 176 192 158)

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

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

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

156 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 24I 24J 24K 24L 24M ··· 24T 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 24 24 24 24 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 10 10 1 1 1 1 2 10 10 2 2 1 1 2 2 10 10 10 10 1 1 1 1 2 2 10 10 10 10 2 ··· 2 1 1 1 1 2 2 10 10 10 10 2 2 2 2 2 ··· 2 1 ··· 1 2 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 D5 D10 D10 C3×D5 C8○D4 C4×D5 C4×D5 C6×D5 C6×D5 C3×C8○D4 D5×C12 D5×C12 D20.3C4 C3×D20.3C4 kernel C3×D20.3C4 D5×C24 C3×C8⋊D5 C3×C4.Dic5 C2×C120 C3×C4○D20 D20.3C4 C3×Dic10 C3×D20 C3×C5⋊D4 C8×D5 C8⋊D5 C4.Dic5 C2×C40 C4○D20 Dic10 D20 C5⋊D4 C2×C24 C24 C2×C12 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 2 4 4 4 2 2 2 4 4 8 2 4 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C3×D20.3C4 in GL2(𝔽241) generated by

 15 0 0 15
,
 238 44 197 163
,
 163 163 44 78
,
 211 0 0 211
G:=sub<GL(2,GF(241))| [15,0,0,15],[238,197,44,163],[163,44,163,78],[211,0,0,211] >;

C3×D20.3C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}._3C_4
% in TeX

G:=Group("C3xD20.3C4");
// GroupNames label

G:=SmallGroup(480,694);
// by ID

G=gap.SmallGroup(480,694);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,142,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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