metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊5S3, C48⋊6D5, C240⋊7C2, C16⋊3D15, D30.3C8, C40.71D6, C8.20D30, C24.76D10, C15⋊10M5(2), Dic15.3C8, C120.89C22, C6.7(C8×D5), C5⋊4(D6.C8), C15⋊3C16⋊7C2, C2.3(C8×D15), C3⋊2(C80⋊C2), C15⋊3C8.6C4, 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
Generators and relations for C80⋊S3
G = < a,b,c | a80=b3=c2=1, ab=ba, cac=a9, cbc=b-1 >
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)
(1 200 116)(2 201 117)(3 202 118)(4 203 119)(5 204 120)(6 205 121)(7 206 122)(8 207 123)(9 208 124)(10 209 125)(11 210 126)(12 211 127)(13 212 128)(14 213 129)(15 214 130)(16 215 131)(17 216 132)(18 217 133)(19 218 134)(20 219 135)(21 220 136)(22 221 137)(23 222 138)(24 223 139)(25 224 140)(26 225 141)(27 226 142)(28 227 143)(29 228 144)(30 229 145)(31 230 146)(32 231 147)(33 232 148)(34 233 149)(35 234 150)(36 235 151)(37 236 152)(38 237 153)(39 238 154)(40 239 155)(41 240 156)(42 161 157)(43 162 158)(44 163 159)(45 164 160)(46 165 81)(47 166 82)(48 167 83)(49 168 84)(50 169 85)(51 170 86)(52 171 87)(53 172 88)(54 173 89)(55 174 90)(56 175 91)(57 176 92)(58 177 93)(59 178 94)(60 179 95)(61 180 96)(62 181 97)(63 182 98)(64 183 99)(65 184 100)(66 185 101)(67 186 102)(68 187 103)(69 188 104)(70 189 105)(71 190 106)(72 191 107)(73 192 108)(74 193 109)(75 194 110)(76 195 111)(77 196 112)(78 197 113)(79 198 114)(80 199 115)
(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)(81 205)(82 214)(83 223)(84 232)(85 161)(86 170)(87 179)(88 188)(89 197)(90 206)(91 215)(92 224)(93 233)(94 162)(95 171)(96 180)(97 189)(98 198)(99 207)(100 216)(101 225)(102 234)(103 163)(104 172)(105 181)(106 190)(107 199)(108 208)(109 217)(110 226)(111 235)(112 164)(113 173)(114 182)(115 191)(116 200)(117 209)(118 218)(119 227)(120 236)(121 165)(122 174)(123 183)(124 192)(125 201)(126 210)(127 219)(128 228)(129 237)(130 166)(131 175)(132 184)(133 193)(134 202)(135 211)(136 220)(137 229)(138 238)(139 167)(140 176)(141 185)(142 194)(143 203)(144 212)(145 221)(146 230)(147 239)(148 168)(149 177)(150 186)(151 195)(152 204)(153 213)(154 222)(155 231)(156 240)(157 169)(158 178)(159 187)(160 196)
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), (1,200,116)(2,201,117)(3,202,118)(4,203,119)(5,204,120)(6,205,121)(7,206,122)(8,207,123)(9,208,124)(10,209,125)(11,210,126)(12,211,127)(13,212,128)(14,213,129)(15,214,130)(16,215,131)(17,216,132)(18,217,133)(19,218,134)(20,219,135)(21,220,136)(22,221,137)(23,222,138)(24,223,139)(25,224,140)(26,225,141)(27,226,142)(28,227,143)(29,228,144)(30,229,145)(31,230,146)(32,231,147)(33,232,148)(34,233,149)(35,234,150)(36,235,151)(37,236,152)(38,237,153)(39,238,154)(40,239,155)(41,240,156)(42,161,157)(43,162,158)(44,163,159)(45,164,160)(46,165,81)(47,166,82)(48,167,83)(49,168,84)(50,169,85)(51,170,86)(52,171,87)(53,172,88)(54,173,89)(55,174,90)(56,175,91)(57,176,92)(58,177,93)(59,178,94)(60,179,95)(61,180,96)(62,181,97)(63,182,98)(64,183,99)(65,184,100)(66,185,101)(67,186,102)(68,187,103)(69,188,104)(70,189,105)(71,190,106)(72,191,107)(73,192,108)(74,193,109)(75,194,110)(76,195,111)(77,196,112)(78,197,113)(79,198,114)(80,199,115), (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)(81,205)(82,214)(83,223)(84,232)(85,161)(86,170)(87,179)(88,188)(89,197)(90,206)(91,215)(92,224)(93,233)(94,162)(95,171)(96,180)(97,189)(98,198)(99,207)(100,216)(101,225)(102,234)(103,163)(104,172)(105,181)(106,190)(107,199)(108,208)(109,217)(110,226)(111,235)(112,164)(113,173)(114,182)(115,191)(116,200)(117,209)(118,218)(119,227)(120,236)(121,165)(122,174)(123,183)(124,192)(125,201)(126,210)(127,219)(128,228)(129,237)(130,166)(131,175)(132,184)(133,193)(134,202)(135,211)(136,220)(137,229)(138,238)(139,167)(140,176)(141,185)(142,194)(143,203)(144,212)(145,221)(146,230)(147,239)(148,168)(149,177)(150,186)(151,195)(152,204)(153,213)(154,222)(155,231)(156,240)(157,169)(158,178)(159,187)(160,196)>;
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), (1,200,116)(2,201,117)(3,202,118)(4,203,119)(5,204,120)(6,205,121)(7,206,122)(8,207,123)(9,208,124)(10,209,125)(11,210,126)(12,211,127)(13,212,128)(14,213,129)(15,214,130)(16,215,131)(17,216,132)(18,217,133)(19,218,134)(20,219,135)(21,220,136)(22,221,137)(23,222,138)(24,223,139)(25,224,140)(26,225,141)(27,226,142)(28,227,143)(29,228,144)(30,229,145)(31,230,146)(32,231,147)(33,232,148)(34,233,149)(35,234,150)(36,235,151)(37,236,152)(38,237,153)(39,238,154)(40,239,155)(41,240,156)(42,161,157)(43,162,158)(44,163,159)(45,164,160)(46,165,81)(47,166,82)(48,167,83)(49,168,84)(50,169,85)(51,170,86)(52,171,87)(53,172,88)(54,173,89)(55,174,90)(56,175,91)(57,176,92)(58,177,93)(59,178,94)(60,179,95)(61,180,96)(62,181,97)(63,182,98)(64,183,99)(65,184,100)(66,185,101)(67,186,102)(68,187,103)(69,188,104)(70,189,105)(71,190,106)(72,191,107)(73,192,108)(74,193,109)(75,194,110)(76,195,111)(77,196,112)(78,197,113)(79,198,114)(80,199,115), (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)(81,205)(82,214)(83,223)(84,232)(85,161)(86,170)(87,179)(88,188)(89,197)(90,206)(91,215)(92,224)(93,233)(94,162)(95,171)(96,180)(97,189)(98,198)(99,207)(100,216)(101,225)(102,234)(103,163)(104,172)(105,181)(106,190)(107,199)(108,208)(109,217)(110,226)(111,235)(112,164)(113,173)(114,182)(115,191)(116,200)(117,209)(118,218)(119,227)(120,236)(121,165)(122,174)(123,183)(124,192)(125,201)(126,210)(127,219)(128,228)(129,237)(130,166)(131,175)(132,184)(133,193)(134,202)(135,211)(136,220)(137,229)(138,238)(139,167)(140,176)(141,185)(142,194)(143,203)(144,212)(145,221)(146,230)(147,239)(148,168)(149,177)(150,186)(151,195)(152,204)(153,213)(154,222)(155,231)(156,240)(157,169)(158,178)(159,187)(160,196) );
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)], [(1,200,116),(2,201,117),(3,202,118),(4,203,119),(5,204,120),(6,205,121),(7,206,122),(8,207,123),(9,208,124),(10,209,125),(11,210,126),(12,211,127),(13,212,128),(14,213,129),(15,214,130),(16,215,131),(17,216,132),(18,217,133),(19,218,134),(20,219,135),(21,220,136),(22,221,137),(23,222,138),(24,223,139),(25,224,140),(26,225,141),(27,226,142),(28,227,143),(29,228,144),(30,229,145),(31,230,146),(32,231,147),(33,232,148),(34,233,149),(35,234,150),(36,235,151),(37,236,152),(38,237,153),(39,238,154),(40,239,155),(41,240,156),(42,161,157),(43,162,158),(44,163,159),(45,164,160),(46,165,81),(47,166,82),(48,167,83),(49,168,84),(50,169,85),(51,170,86),(52,171,87),(53,172,88),(54,173,89),(55,174,90),(56,175,91),(57,176,92),(58,177,93),(59,178,94),(60,179,95),(61,180,96),(62,181,97),(63,182,98),(64,183,99),(65,184,100),(66,185,101),(67,186,102),(68,187,103),(69,188,104),(70,189,105),(71,190,106),(72,191,107),(73,192,108),(74,193,109),(75,194,110),(76,195,111),(77,196,112),(78,197,113),(79,198,114),(80,199,115)], [(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),(81,205),(82,214),(83,223),(84,232),(85,161),(86,170),(87,179),(88,188),(89,197),(90,206),(91,215),(92,224),(93,233),(94,162),(95,171),(96,180),(97,189),(98,198),(99,207),(100,216),(101,225),(102,234),(103,163),(104,172),(105,181),(106,190),(107,199),(108,208),(109,217),(110,226),(111,235),(112,164),(113,173),(114,182),(115,191),(116,200),(117,209),(118,218),(119,227),(120,236),(121,165),(122,174),(123,183),(124,192),(125,201),(126,210),(127,219),(128,228),(129,237),(130,166),(131,175),(132,184),(133,193),(134,202),(135,211),(136,220),(137,229),(138,238),(139,167),(140,176),(141,185),(142,194),(143,203),(144,212),(145,221),(146,230),(147,239),(148,168),(149,177),(150,186),(151,195),(152,204),(153,213),(154,222),(155,231),(156,240),(157,169),(158,178),(159,187),(160,196)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 30 | 2 | 1 | 1 | 30 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D5 | D6 | D10 | C4×S3 | D15 | M5(2) | C4×D5 | S3×C8 | D30 | C8×D5 | D6.C8 | C4×D15 | C80⋊C2 | C8×D15 | C80⋊S3 |
| kernel | C80⋊S3 | C15⋊3C16 | C240 | C8×D15 | C15⋊3C8 | C4×D15 | Dic15 | D30 | C80 | C48 | C40 | C24 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C80⋊S3 ►in GL4(𝔽241) generated by
| 140 | 164 | 0 | 0 |
| 239 | 101 | 0 | 0 |
| 0 | 0 | 8 | 233 |
| 0 | 0 | 183 | 66 |
| 9 | 49 | 0 | 0 |
| 67 | 231 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 157 | 240 | 0 | 0 |
| 0 | 0 | 173 | 148 |
| 0 | 0 | 143 | 68 |
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