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## G = C3×C80⋊C2order 480 = 25·3·5

### Direct product of C3 and C80⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C80⋊C2
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — D5×C24 — C3×C80⋊C2
 Lower central C5 — C10 — C3×C80⋊C2
 Upper central C1 — C24 — C48

Generators and relations for C3×C80⋊C2
G = < a,b,c | a3=b80=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C3×C80⋊C2
On 240 points
Generators in S240
(1 217 111)(2 218 112)(3 219 113)(4 220 114)(5 221 115)(6 222 116)(7 223 117)(8 224 118)(9 225 119)(10 226 120)(11 227 121)(12 228 122)(13 229 123)(14 230 124)(15 231 125)(16 232 126)(17 233 127)(18 234 128)(19 235 129)(20 236 130)(21 237 131)(22 238 132)(23 239 133)(24 240 134)(25 161 135)(26 162 136)(27 163 137)(28 164 138)(29 165 139)(30 166 140)(31 167 141)(32 168 142)(33 169 143)(34 170 144)(35 171 145)(36 172 146)(37 173 147)(38 174 148)(39 175 149)(40 176 150)(41 177 151)(42 178 152)(43 179 153)(44 180 154)(45 181 155)(46 182 156)(47 183 157)(48 184 158)(49 185 159)(50 186 160)(51 187 81)(52 188 82)(53 189 83)(54 190 84)(55 191 85)(56 192 86)(57 193 87)(58 194 88)(59 195 89)(60 196 90)(61 197 91)(62 198 92)(63 199 93)(64 200 94)(65 201 95)(66 202 96)(67 203 97)(68 204 98)(69 205 99)(70 206 100)(71 207 101)(72 208 102)(73 209 103)(74 210 104)(75 211 105)(76 212 106)(77 213 107)(78 214 108)(79 215 109)(80 216 110)
(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)
(2 10)(3 19)(4 28)(5 37)(6 46)(7 55)(8 64)(9 73)(12 20)(13 29)(14 38)(15 47)(16 56)(17 65)(18 74)(22 30)(23 39)(24 48)(25 57)(26 66)(27 75)(32 40)(33 49)(34 58)(35 67)(36 76)(42 50)(43 59)(44 68)(45 77)(52 60)(53 69)(54 78)(62 70)(63 79)(72 80)(82 90)(83 99)(84 108)(85 117)(86 126)(87 135)(88 144)(89 153)(92 100)(93 109)(94 118)(95 127)(96 136)(97 145)(98 154)(102 110)(103 119)(104 128)(105 137)(106 146)(107 155)(112 120)(113 129)(114 138)(115 147)(116 156)(122 130)(123 139)(124 148)(125 157)(132 140)(133 149)(134 158)(142 150)(143 159)(152 160)(161 193)(162 202)(163 211)(164 220)(165 229)(166 238)(168 176)(169 185)(170 194)(171 203)(172 212)(173 221)(174 230)(175 239)(178 186)(179 195)(180 204)(181 213)(182 222)(183 231)(184 240)(188 196)(189 205)(190 214)(191 223)(192 232)(198 206)(199 215)(200 224)(201 233)(208 216)(209 225)(210 234)(218 226)(219 235)(228 236)

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

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

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

156 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 10A 10B 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 24A ··· 24H 24I 24J 24K 24L 30A 30B 30C 30D 40A ··· 40H 48A ··· 48H 48I ··· 48P 60A ··· 60H 80A ··· 80P 120A ··· 120P 240A ··· 240AF order 1 2 2 3 3 4 4 4 5 5 6 6 6 6 8 8 8 8 8 8 10 10 12 12 12 12 12 12 15 15 15 15 16 16 16 16 16 16 16 16 20 20 20 20 24 ··· 24 24 24 24 24 30 30 30 30 40 ··· 40 48 ··· 48 48 ··· 48 60 ··· 60 80 ··· 80 120 ··· 120 240 ··· 240 size 1 1 10 1 1 1 1 10 2 2 1 1 10 10 1 1 1 1 10 10 2 2 1 1 1 1 10 10 2 2 2 2 2 2 2 2 10 10 10 10 2 2 2 2 1 ··· 1 10 10 10 10 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C8 C12 C12 C24 C24 D5 D10 C3×D5 M5(2) C4×D5 C6×D5 C8×D5 C3×M5(2) D5×C12 C80⋊C2 D5×C24 C3×C80⋊C2 kernel C3×C80⋊C2 C3×C5⋊2C16 C240 D5×C24 C80⋊C2 C3×C5⋊2C8 D5×C12 C5⋊2C16 C80 C8×D5 C3×Dic5 C6×D5 C5⋊2C8 C4×D5 Dic5 D10 C48 C24 C16 C15 C12 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 2 2 4 4 4 4 8 8 8 16 16 32

Matrix representation of C3×C80⋊C2 in GL2(𝔽241) generated by

 15 0 0 15
,
 189 188 1 136
,
 189 53 190 52
G:=sub<GL(2,GF(241))| [15,0,0,15],[189,1,188,136],[189,190,53,52] >;

C3×C80⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{80}\rtimes C_2
% in TeX

G:=Group("C3xC80:C2");
// GroupNames label

G:=SmallGroup(480,76);
// by ID

G=gap.SmallGroup(480,76);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,701,92,80,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^3=b^80=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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