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G = C3⋊D80order 480 = 25·3·5

The semidirect product of C3 and D80 acting via D80/D40=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C32D80, C152D16, D401S3, C6.6D40, C30.2D8, D1203C2, C60.24D4, C40.40D6, C24.5D10, C12.1D20, C120.2C22, C3⋊C161D5, C8.7(S3×D5), (C3×D40)⋊1C2, C51(C3⋊D16), C10.1(D4⋊S3), C2.4(C3⋊D40), C4.1(C3⋊D20), C20.49(C3⋊D4), (C5×C3⋊C16)⋊1C2, SmallGroup(480,14)

Series: Derived Chief Lower central Upper central

C1C120 — C3⋊D80
C1C5C15C30C60C120C3×D40 — C3⋊D80
C15C30C60C120 — C3⋊D80
C1C2C4C8

Generators and relations for C3⋊D80
 G = < a,b,c | a3=b80=c2=1, bab-1=cac=a-1, cbc=b-1 >

40C2
120C2
20C22
60C22
40C6
40S3
8D5
24D5
10D4
30D4
20D6
20C2×C6
4D10
12D10
8D15
8C3×D5
3C16
5D8
15D8
10C3×D4
10D12
2D20
6D20
4D30
4C6×D5
15D16
5C3×D8
5D24
3C80
3D40
2C3×D20
2D60
5C3⋊D16
3D80

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

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

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

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

66 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C8A8B10A10B 12 15A15B16A16B16C16D20A20B20C20D24A24B30A30B40A···40H60A60B60C60D80A···80P120A···120H
order1222345566688101012151516161616202020202424303040···406060606080···80120···120
size114012022222404022224446666222244442···244446···64···4

66 irreducible representations

dim111122222222222444444
type++++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D16D20D40D80D4⋊S3S3×D5C3⋊D16C3⋊D20C3⋊D40C3⋊D80
kernelC3⋊D80C5×C3⋊C16C3×D40D120D40C60C3⋊C16C40C30C24C20C15C12C6C3C10C8C5C4C2C1
# reps1111112122244816122248

Matrix representation of C3⋊D80 in GL6(𝔽241)

100000
010000
001000
000100
00002401
00002400
,
227420000
1992120000
0023713600
001565800
000010170
0000171140
,
119410000
1191220000
002411400
0010721700
00000240
00002400

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[227,199,0,0,0,0,42,212,0,0,0,0,0,0,237,156,0,0,0,0,136,58,0,0,0,0,0,0,101,171,0,0,0,0,70,140],[119,119,0,0,0,0,41,122,0,0,0,0,0,0,24,107,0,0,0,0,114,217,0,0,0,0,0,0,0,240,0,0,0,0,240,0] >;

C3⋊D80 in GAP, Magma, Sage, TeX

C_3\rtimes D_{80}
% in TeX

G:=Group("C3:D80");
// GroupNames label

G:=SmallGroup(480,14);
// by ID

G=gap.SmallGroup(480,14);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,254,142,675,80,1356,18822]);
// Polycyclic

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

Export

Subgroup lattice of C3⋊D80 in TeX

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