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G = C80⋊S3order 480 = 25·3·5

5th semidirect product of C80 and S3 acting via S3/C3=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C805S3, C486D5, C2407C2, C163D15, D30.3C8, C40.71D6, C8.20D30, C24.76D10, C1510M5(2), Dic15.3C8, C120.89C22, C6.7(C8×D5), C54(D6.C8), C153C167C2, C2.3(C8×D15), C32(C80⋊C2), C153C8.6C4, C10.16(S3×C8), C20.87(C4×S3), C30.44(C2×C8), (C4×D15).8C4, (C8×D15).3C2, C4.17(C4×D15), C12.55(C4×D5), C60.192(C2×C4), SmallGroup(480,158)

Series: Derived Chief Lower central Upper central

C1C30 — C80⋊S3
C1C5C15C30C60C120C8×D15 — C80⋊S3
C15C30 — C80⋊S3
C1C8C16

Generators and relations for C80⋊S3
 G = < a,b,c | a80=b3=c2=1, ab=ba, cac=a9, cbc=b-1 >

30C2
15C22
15C4
10S3
6D5
15C2×C4
15C8
5D6
5Dic3
3Dic5
3D10
2D15
15C16
15C2×C8
5C3⋊C8
5C4×S3
3C52C8
3C4×D5
15M5(2)
5C3⋊C16
5S3×C8
3C8×D5
3C52C16
5D6.C8
3C80⋊C2

Smallest permutation representation of C80⋊S3
On 240 points
Generators in S240
(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)
(1 152 222)(2 153 223)(3 154 224)(4 155 225)(5 156 226)(6 157 227)(7 158 228)(8 159 229)(9 160 230)(10 81 231)(11 82 232)(12 83 233)(13 84 234)(14 85 235)(15 86 236)(16 87 237)(17 88 238)(18 89 239)(19 90 240)(20 91 161)(21 92 162)(22 93 163)(23 94 164)(24 95 165)(25 96 166)(26 97 167)(27 98 168)(28 99 169)(29 100 170)(30 101 171)(31 102 172)(32 103 173)(33 104 174)(34 105 175)(35 106 176)(36 107 177)(37 108 178)(38 109 179)(39 110 180)(40 111 181)(41 112 182)(42 113 183)(43 114 184)(44 115 185)(45 116 186)(46 117 187)(47 118 188)(48 119 189)(49 120 190)(50 121 191)(51 122 192)(52 123 193)(53 124 194)(54 125 195)(55 126 196)(56 127 197)(57 128 198)(58 129 199)(59 130 200)(60 131 201)(61 132 202)(62 133 203)(63 134 204)(64 135 205)(65 136 206)(66 137 207)(67 138 208)(68 139 209)(69 140 210)(70 141 211)(71 142 212)(72 143 213)(73 144 214)(74 145 215)(75 146 216)(76 147 217)(77 148 218)(78 149 219)(79 150 220)(80 151 221)
(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)(81 223)(82 232)(83 161)(84 170)(85 179)(86 188)(87 197)(88 206)(89 215)(90 224)(91 233)(92 162)(93 171)(94 180)(95 189)(96 198)(97 207)(98 216)(99 225)(100 234)(101 163)(102 172)(103 181)(104 190)(105 199)(106 208)(107 217)(108 226)(109 235)(110 164)(111 173)(112 182)(113 191)(114 200)(115 209)(116 218)(117 227)(118 236)(119 165)(120 174)(121 183)(122 192)(123 201)(124 210)(125 219)(126 228)(127 237)(128 166)(129 175)(130 184)(131 193)(132 202)(133 211)(134 220)(135 229)(136 238)(137 167)(138 176)(139 185)(140 194)(141 203)(142 212)(143 221)(144 230)(145 239)(146 168)(147 177)(148 186)(149 195)(150 204)(151 213)(152 222)(153 231)(154 240)(155 169)(156 178)(157 187)(158 196)(159 205)(160 214)

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

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

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

132 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B 6 8A8B8C8D8E8F10A10B12A12B15A15B15C15D16A16B16C16D16E16F16G16H20A20B20C20D24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order12234445568888881010121215151515161616161616161620202020242424243030303040···4048···4860···6080···80120···120240···240
size11302113022211113030222222222222303030302222222222222···22···22···22···22···22···2

132 irreducible representations

dim111111112222222222222222
type++++++++++
imageC1C2C2C2C4C4C8C8S3D5D6D10C4×S3D15M5(2)C4×D5S3×C8D30C8×D5D6.C8C4×D15C80⋊C2C8×D15C80⋊S3
kernelC80⋊S3C153C16C240C8×D15C153C8C4×D15Dic15D30C80C48C40C24C20C16C15C12C10C8C6C5C4C3C2C1
# reps111122441212244444888161632

Matrix representation of C80⋊S3 in GL4(𝔽241) generated by

14016400
23910100
008233
0018366
,
94900
6723100
0010
0001
,
1000
15724000
00173148
0014368
G:=sub<GL(4,GF(241))| [140,239,0,0,164,101,0,0,0,0,8,183,0,0,233,66],[9,67,0,0,49,231,0,0,0,0,1,0,0,0,0,1],[1,157,0,0,0,240,0,0,0,0,173,143,0,0,148,68] >;

C80⋊S3 in GAP, Magma, Sage, TeX

C_{80}\rtimes S_3
% in TeX

G:=Group("C80:S3");
// GroupNames label

G:=SmallGroup(480,158);
// by ID

G=gap.SmallGroup(480,158);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,36,58,80,2693,18822]);
// Polycyclic

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

Export

Subgroup lattice of C80⋊S3 in TeX

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