direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C39, C5⋊C78, C65⋊7C6, C15⋊2C26, C195⋊6C2, SmallGroup(390,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C39 |
Generators and relations for D5×C39
G = < a,b,c | a39=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C39
►On 195 points
Generators in S195
(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 76 125 89 170)(2 77 126 90 171)(3 78 127 91 172)(4 40 128 92 173)(5 41 129 93 174)(6 42 130 94 175)(7 43 131 95 176)(8 44 132 96 177)(9 45 133 97 178)(10 46 134 98 179)(11 47 135 99 180)(12 48 136 100 181)(13 49 137 101 182)(14 50 138 102 183)(15 51 139 103 184)(16 52 140 104 185)(17 53 141 105 186)(18 54 142 106 187)(19 55 143 107 188)(20 56 144 108 189)(21 57 145 109 190)(22 58 146 110 191)(23 59 147 111 192)(24 60 148 112 193)(25 61 149 113 194)(26 62 150 114 195)(27 63 151 115 157)(28 64 152 116 158)(29 65 153 117 159)(30 66 154 79 160)(31 67 155 80 161)(32 68 156 81 162)(33 69 118 82 163)(34 70 119 83 164)(35 71 120 84 165)(36 72 121 85 166)(37 73 122 86 167)(38 74 123 87 168)(39 75 124 88 169)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 193)(25 194)(26 195)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)
G:=sub<Sym(195)| (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), (1,76,125,89,170)(2,77,126,90,171)(3,78,127,91,172)(4,40,128,92,173)(5,41,129,93,174)(6,42,130,94,175)(7,43,131,95,176)(8,44,132,96,177)(9,45,133,97,178)(10,46,134,98,179)(11,47,135,99,180)(12,48,136,100,181)(13,49,137,101,182)(14,50,138,102,183)(15,51,139,103,184)(16,52,140,104,185)(17,53,141,105,186)(18,54,142,106,187)(19,55,143,107,188)(20,56,144,108,189)(21,57,145,109,190)(22,58,146,110,191)(23,59,147,111,192)(24,60,148,112,193)(25,61,149,113,194)(26,62,150,114,195)(27,63,151,115,157)(28,64,152,116,158)(29,65,153,117,159)(30,66,154,79,160)(31,67,155,80,161)(32,68,156,81,162)(33,69,118,82,163)(34,70,119,83,164)(35,71,120,84,165)(36,72,121,85,166)(37,73,122,86,167)(38,74,123,87,168)(39,75,124,88,169), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)>;
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), (1,76,125,89,170)(2,77,126,90,171)(3,78,127,91,172)(4,40,128,92,173)(5,41,129,93,174)(6,42,130,94,175)(7,43,131,95,176)(8,44,132,96,177)(9,45,133,97,178)(10,46,134,98,179)(11,47,135,99,180)(12,48,136,100,181)(13,49,137,101,182)(14,50,138,102,183)(15,51,139,103,184)(16,52,140,104,185)(17,53,141,105,186)(18,54,142,106,187)(19,55,143,107,188)(20,56,144,108,189)(21,57,145,109,190)(22,58,146,110,191)(23,59,147,111,192)(24,60,148,112,193)(25,61,149,113,194)(26,62,150,114,195)(27,63,151,115,157)(28,64,152,116,158)(29,65,153,117,159)(30,66,154,79,160)(31,67,155,80,161)(32,68,156,81,162)(33,69,118,82,163)(34,70,119,83,164)(35,71,120,84,165)(36,72,121,85,166)(37,73,122,86,167)(38,74,123,87,168)(39,75,124,88,169), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91) );
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)], [(1,76,125,89,170),(2,77,126,90,171),(3,78,127,91,172),(4,40,128,92,173),(5,41,129,93,174),(6,42,130,94,175),(7,43,131,95,176),(8,44,132,96,177),(9,45,133,97,178),(10,46,134,98,179),(11,47,135,99,180),(12,48,136,100,181),(13,49,137,101,182),(14,50,138,102,183),(15,51,139,103,184),(16,52,140,104,185),(17,53,141,105,186),(18,54,142,106,187),(19,55,143,107,188),(20,56,144,108,189),(21,57,145,109,190),(22,58,146,110,191),(23,59,147,111,192),(24,60,148,112,193),(25,61,149,113,194),(26,62,150,114,195),(27,63,151,115,157),(28,64,152,116,158),(29,65,153,117,159),(30,66,154,79,160),(31,67,155,80,161),(32,68,156,81,162),(33,69,118,82,163),(34,70,119,83,164),(35,71,120,84,165),(36,72,121,85,166),(37,73,122,86,167),(38,74,123,87,168),(39,75,124,88,169)], [(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,181),(13,182),(14,183),(15,184),(16,185),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,193),(25,194),(26,195),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,116),(65,117),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91)]])
156 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 13A | ··· | 13L | 15A | 15B | 15C | 15D | 26A | ··· | 26L | 39A | ··· | 39X | 65A | ··· | 65X | 78A | ··· | 78X | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 13 | ··· | 13 | 15 | 15 | 15 | 15 | 26 | ··· | 26 | 39 | ··· | 39 | 65 | ··· | 65 | 78 | ··· | 78 | 195 | ··· | 195 |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 | D5 | C3×D5 | D5×C13 | D5×C39 |
| kernel | D5×C39 | C195 | D5×C13 | C65 | C3×D5 | C15 | D5 | C5 | C39 | C13 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 | 2 | 4 | 24 | 48 |
Matrix representation of D5×C39 ►in GL3(𝔽1171) generated by
| 420 | 0 | 0 |
| 0 | 370 | 0 |
| 0 | 0 | 370 |
| 1 | 0 | 0 |
| 0 | 113 | 1 |
| 0 | 1170 | 0 |
| 1170 | 0 | 0 |
| 0 | 1 | 113 |
| 0 | 0 | 1170 |
G:=sub<GL(3,GF(1171))| [420,0,0,0,370,0,0,0,370],[1,0,0,0,113,1170,0,1,0],[1170,0,0,0,1,0,0,113,1170] >;
D5×C39 in GAP, Magma, Sage, TeX
D_5\times C_{39} % in TeX
G:=Group("D5xC39"); // GroupNames label
G:=SmallGroup(390,6);
// by ID
G=gap.SmallGroup(390,6);
# by ID
G:=PCGroup([4,-2,-3,-13,-5,4995]);
// Polycyclic
G:=Group<a,b,c|a^39=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export