direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D39, C65⋊2S3, C39⋊1C10, C195⋊3C2, C15⋊3D13, C13⋊(C5×S3), C3⋊(C5×D13), SmallGroup(390,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C5×D39 |
Generators and relations for C5×D39
G = < a,b,c | a5=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D39
►On 195 points
Generators in S195
(1 172 139 92 63)(2 173 140 93 64)(3 174 141 94 65)(4 175 142 95 66)(5 176 143 96 67)(6 177 144 97 68)(7 178 145 98 69)(8 179 146 99 70)(9 180 147 100 71)(10 181 148 101 72)(11 182 149 102 73)(12 183 150 103 74)(13 184 151 104 75)(14 185 152 105 76)(15 186 153 106 77)(16 187 154 107 78)(17 188 155 108 40)(18 189 156 109 41)(19 190 118 110 42)(20 191 119 111 43)(21 192 120 112 44)(22 193 121 113 45)(23 194 122 114 46)(24 195 123 115 47)(25 157 124 116 48)(26 158 125 117 49)(27 159 126 79 50)(28 160 127 80 51)(29 161 128 81 52)(30 162 129 82 53)(31 163 130 83 54)(32 164 131 84 55)(33 165 132 85 56)(34 166 133 86 57)(35 167 134 87 58)(36 168 135 88 59)(37 169 136 89 60)(38 170 137 90 61)(39 171 138 91 62)
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 46)(41 45)(42 44)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 120)(121 156)(122 155)(123 154)(124 153)(125 152)(126 151)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(157 186)(158 185)(159 184)(160 183)(161 182)(162 181)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(187 195)(188 194)(189 193)(190 192)
G:=sub<Sym(195)| (1,172,139,92,63)(2,173,140,93,64)(3,174,141,94,65)(4,175,142,95,66)(5,176,143,96,67)(6,177,144,97,68)(7,178,145,98,69)(8,179,146,99,70)(9,180,147,100,71)(10,181,148,101,72)(11,182,149,102,73)(12,183,150,103,74)(13,184,151,104,75)(14,185,152,105,76)(15,186,153,106,77)(16,187,154,107,78)(17,188,155,108,40)(18,189,156,109,41)(19,190,118,110,42)(20,191,119,111,43)(21,192,120,112,44)(22,193,121,113,45)(23,194,122,114,46)(24,195,123,115,47)(25,157,124,116,48)(26,158,125,117,49)(27,159,126,79,50)(28,160,127,80,51)(29,161,128,81,52)(30,162,129,82,53)(31,163,130,83,54)(32,164,131,84,55)(33,165,132,85,56)(34,166,133,86,57)(35,167,134,87,58)(36,168,135,88,59)(37,169,136,89,60)(38,170,137,90,61)(39,171,138,91,62), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,46)(41,45)(42,44)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,120)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(187,195)(188,194)(189,193)(190,192)>;
G:=Group( (1,172,139,92,63)(2,173,140,93,64)(3,174,141,94,65)(4,175,142,95,66)(5,176,143,96,67)(6,177,144,97,68)(7,178,145,98,69)(8,179,146,99,70)(9,180,147,100,71)(10,181,148,101,72)(11,182,149,102,73)(12,183,150,103,74)(13,184,151,104,75)(14,185,152,105,76)(15,186,153,106,77)(16,187,154,107,78)(17,188,155,108,40)(18,189,156,109,41)(19,190,118,110,42)(20,191,119,111,43)(21,192,120,112,44)(22,193,121,113,45)(23,194,122,114,46)(24,195,123,115,47)(25,157,124,116,48)(26,158,125,117,49)(27,159,126,79,50)(28,160,127,80,51)(29,161,128,81,52)(30,162,129,82,53)(31,163,130,83,54)(32,164,131,84,55)(33,165,132,85,56)(34,166,133,86,57)(35,167,134,87,58)(36,168,135,88,59)(37,169,136,89,60)(38,170,137,90,61)(39,171,138,91,62), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,46)(41,45)(42,44)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,120)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(187,195)(188,194)(189,193)(190,192) );
G=PermutationGroup([[(1,172,139,92,63),(2,173,140,93,64),(3,174,141,94,65),(4,175,142,95,66),(5,176,143,96,67),(6,177,144,97,68),(7,178,145,98,69),(8,179,146,99,70),(9,180,147,100,71),(10,181,148,101,72),(11,182,149,102,73),(12,183,150,103,74),(13,184,151,104,75),(14,185,152,105,76),(15,186,153,106,77),(16,187,154,107,78),(17,188,155,108,40),(18,189,156,109,41),(19,190,118,110,42),(20,191,119,111,43),(21,192,120,112,44),(22,193,121,113,45),(23,194,122,114,46),(24,195,123,115,47),(25,157,124,116,48),(26,158,125,117,49),(27,159,126,79,50),(28,160,127,80,51),(29,161,128,81,52),(30,162,129,82,53),(31,163,130,83,54),(32,164,131,84,55),(33,165,132,85,56),(34,166,133,86,57),(35,167,134,87,58),(36,168,135,88,59),(37,169,136,89,60),(38,170,137,90,61),(39,171,138,91,62)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,46),(41,45),(42,44),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(118,120),(121,156),(122,155),(123,154),(124,153),(125,152),(126,151),(127,150),(128,149),(129,148),(130,147),(131,146),(132,145),(133,144),(134,143),(135,142),(136,141),(137,140),(138,139),(157,186),(158,185),(159,184),(160,183),(161,182),(162,181),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(187,195),(188,194),(189,193),(190,192)]])
105 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 13A | ··· | 13F | 15A | 15B | 15C | 15D | 39A | ··· | 39L | 65A | ··· | 65X | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 13 | ··· | 13 | 15 | 15 | 15 | 15 | 39 | ··· | 39 | 65 | ··· | 65 | 195 | ··· | 195 |
| size | 1 | 39 | 2 | 1 | 1 | 1 | 1 | 39 | 39 | 39 | 39 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | D13 | C5×S3 | D39 | C5×D13 | C5×D39 |
| kernel | C5×D39 | C195 | D39 | C39 | C65 | C15 | C13 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 6 | 4 | 12 | 24 | 48 |
Matrix representation of C5×D39 ►in GL2(𝔽1171) generated by
| 216 | 0 |
| 0 | 216 |
| 1029 | 1037 |
| 134 | 580 |
| 398 | 834 |
| 682 | 773 |
G:=sub<GL(2,GF(1171))| [216,0,0,216],[1029,134,1037,580],[398,682,834,773] >;
C5×D39 in GAP, Magma, Sage, TeX
C_5\times D_{39} % in TeX
G:=Group("C5xD39"); // GroupNames label
G:=SmallGroup(390,9);
// by ID
G=gap.SmallGroup(390,9);
# by ID
G:=PCGroup([4,-2,-5,-3,-13,242,5763]);
// Polycyclic
G:=Group<a,b,c|a^5=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export