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G = C3×D80order 480 = 25·3·5

Direct product of C3 and D80

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

Aliases: C3×D80, C801C6, C483D5, C155D16, C2403C2, D401C6, C6.13D40, C30.27D8, C60.169D4, C12.39D20, C24.70D10, C120.83C22, C51(C3×D16), C161(C3×D5), (C3×D40)⋊5C2, C2.3(C3×D40), C8.13(C6×D5), C4.1(C3×D20), C10.1(C3×D8), C40.14(C2×C6), C20.24(C3×D4), SmallGroup(480,77)

Series: Derived Chief Lower central Upper central

C1C40 — C3×D80
C1C5C10C20C40C120C3×D40 — C3×D80
C5C10C20C40 — C3×D80
C1C6C12C24C48

Generators and relations for C3×D80
 G = < a,b,c | a3=b80=c2=1, ab=ba, ac=ca, cbc=b-1 >

40C2
40C2
20C22
20C22
40C6
40C6
8D5
8D5
10D4
10D4
20C2×C6
20C2×C6
4D10
4D10
8C3×D5
8C3×D5
5D8
5D8
10C3×D4
10C3×D4
2D20
2D20
4C6×D5
4C6×D5
5D16
5C3×D8
5C3×D8
2C3×D20
2C3×D20
5C3×D16

Smallest permutation representation of C3×D80
On 240 points
Generators in S240
(1 194 96)(2 195 97)(3 196 98)(4 197 99)(5 198 100)(6 199 101)(7 200 102)(8 201 103)(9 202 104)(10 203 105)(11 204 106)(12 205 107)(13 206 108)(14 207 109)(15 208 110)(16 209 111)(17 210 112)(18 211 113)(19 212 114)(20 213 115)(21 214 116)(22 215 117)(23 216 118)(24 217 119)(25 218 120)(26 219 121)(27 220 122)(28 221 123)(29 222 124)(30 223 125)(31 224 126)(32 225 127)(33 226 128)(34 227 129)(35 228 130)(36 229 131)(37 230 132)(38 231 133)(39 232 134)(40 233 135)(41 234 136)(42 235 137)(43 236 138)(44 237 139)(45 238 140)(46 239 141)(47 240 142)(48 161 143)(49 162 144)(50 163 145)(51 164 146)(52 165 147)(53 166 148)(54 167 149)(55 168 150)(56 169 151)(57 170 152)(58 171 153)(59 172 154)(60 173 155)(61 174 156)(62 175 157)(63 176 158)(64 177 159)(65 178 160)(66 179 81)(67 180 82)(68 181 83)(69 182 84)(70 183 85)(71 184 86)(72 185 87)(73 186 88)(74 187 89)(75 188 90)(76 189 91)(77 190 92)(78 191 93)(79 192 94)(80 193 95)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(81 125)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(126 160)(127 159)(128 158)(129 157)(130 156)(131 155)(132 154)(133 153)(134 152)(135 151)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(162 240)(163 239)(164 238)(165 237)(166 236)(167 235)(168 234)(169 233)(170 232)(171 231)(172 230)(173 229)(174 228)(175 227)(176 226)(177 225)(178 224)(179 223)(180 222)(181 221)(182 220)(183 219)(184 218)(185 217)(186 216)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)

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

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

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

129 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B6A6B6C6D6E6F8A8B10A10B12A12B15A15B15C15D16A16B16C16D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order1222334556666668810101212151515151616161620202020242424243030303040···4048···4860···6080···80120···120240···240
size114040112221140404040222222222222222222222222222···22···22···22···22···22···2

129 irreducible representations

dim1111112222222222222222
type+++++++++++
imageC1C2C2C3C6C6D4D5D8D10C3×D4C3×D5D16D20C3×D8C6×D5D40C3×D16C3×D20D80C3×D40C3×D80
kernelC3×D80C240C3×D40D80C80D40C60C48C30C24C20C16C15C12C10C8C6C5C4C3C2C1
# reps1122241222244444888161632

Matrix representation of C3×D80 in GL2(𝔽241) generated by

150
015
,
160139
10260
,
27185
13214
G:=sub<GL(2,GF(241))| [15,0,0,15],[160,102,139,60],[27,13,185,214] >;

C3×D80 in GAP, Magma, Sage, TeX

C_3\times D_{80}
% in TeX

G:=Group("C3xD80");
// GroupNames label

G:=SmallGroup(480,77);
// by ID

G=gap.SmallGroup(480,77);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,260,1011,192,2524,102,18822]);
// 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^-1>;
// generators/relations

Export

Subgroup lattice of C3×D80 in TeX

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