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G = C3×C80⋊C2order 480 = 25·3·5

Direct product of C3 and C80⋊C2

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

Aliases: C3×C80⋊C2, C805C6, C487D5, C24011C2, D10.1C24, C24.81D10, C1511M5(2), Dic5.1C24, C120.99C22, C163(C3×D5), C52C164C6, (C8×D5).2C6, (C6×D5).3C8, C8.20(C6×D5), C6.17(C8×D5), C2.3(D5×C24), C53(C3×M5(2)), C52C8.2C12, C30.49(C2×C8), C40.20(C2×C6), (D5×C12).8C4, (C4×D5).2C12, (D5×C24).9C2, C4.17(D5×C12), C12.87(C4×D5), C60.209(C2×C4), C10.11(C2×C24), C20.43(C2×C12), (C3×Dic5).3C8, (C3×C52C8).7C4, (C3×C52C16)⋊11C2, SmallGroup(480,76)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C80⋊C2
C1C5C10C20C40C120D5×C24 — C3×C80⋊C2
C5C10 — C3×C80⋊C2
C1C24C48

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

10C2
5C22
5C4
10C6
2D5
5C2×C4
5C8
5C2×C6
5C12
2C3×D5
5C16
5C2×C8
5C24
5C2×C12
5M5(2)
5C48
5C2×C24
5C3×M5(2)

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 2A2B3A3B4A4B4C5A5B6A6B6C6D8A8B8C8D8E8F10A10B12A12B12C12D12E12F15A15B15C15D16A16B16C16D16E16F16G16H20A20B20C20D24A···24H24I24J24K24L30A30B30C30D40A···40H48A···48H48I···48P60A···60H80A···80P120A···120P240A···240AF
order1223344455666688888810101212121212121515151516161616161616162020202024···24242424243030303040···4048···4848···4860···6080···80120···120240···240
size111011111022111010111110102211111010222222221010101022221···11010101022222···22···210···102···22···22···22···2

156 irreducible representations

dim1111111111111111222222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C8C12C12C24C24D5D10C3×D5M5(2)C4×D5C6×D5C8×D5C3×M5(2)D5×C12C80⋊C2D5×C24C3×C80⋊C2
kernelC3×C80⋊C2C3×C52C16C240D5×C24C80⋊C2C3×C52C8D5×C12C52C16C80C8×D5C3×Dic5C6×D5C52C8C4×D5Dic5D10C48C24C16C15C12C8C6C5C4C3C2C1
# reps1111222222444488224444888161632

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

150
015
,
189188
1136
,
18953
19052
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

Export

Subgroup lattice of C3×C80⋊C2 in TeX

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