direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C31, C5⋊C62, C155⋊3C2, SmallGroup(310,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C31 |
Generators and relations for D5×C31
G = < a,b,c | a31=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C31
►On 155 points
Generators in S155
(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)
(1 56 134 64 123)(2 57 135 65 124)(3 58 136 66 94)(4 59 137 67 95)(5 60 138 68 96)(6 61 139 69 97)(7 62 140 70 98)(8 32 141 71 99)(9 33 142 72 100)(10 34 143 73 101)(11 35 144 74 102)(12 36 145 75 103)(13 37 146 76 104)(14 38 147 77 105)(15 39 148 78 106)(16 40 149 79 107)(17 41 150 80 108)(18 42 151 81 109)(19 43 152 82 110)(20 44 153 83 111)(21 45 154 84 112)(22 46 155 85 113)(23 47 125 86 114)(24 48 126 87 115)(25 49 127 88 116)(26 50 128 89 117)(27 51 129 90 118)(28 52 130 91 119)(29 53 131 92 120)(30 54 132 93 121)(31 55 133 63 122)
(1 123)(2 124)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)
G:=sub<Sym(155)| (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), (1,56,134,64,123)(2,57,135,65,124)(3,58,136,66,94)(4,59,137,67,95)(5,60,138,68,96)(6,61,139,69,97)(7,62,140,70,98)(8,32,141,71,99)(9,33,142,72,100)(10,34,143,73,101)(11,35,144,74,102)(12,36,145,75,103)(13,37,146,76,104)(14,38,147,77,105)(15,39,148,78,106)(16,40,149,79,107)(17,41,150,80,108)(18,42,151,81,109)(19,43,152,82,110)(20,44,153,83,111)(21,45,154,84,112)(22,46,155,85,113)(23,47,125,86,114)(24,48,126,87,115)(25,49,127,88,116)(26,50,128,89,117)(27,51,129,90,118)(28,52,130,91,119)(29,53,131,92,120)(30,54,132,93,121)(31,55,133,63,122), (1,123)(2,124)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)>;
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), (1,56,134,64,123)(2,57,135,65,124)(3,58,136,66,94)(4,59,137,67,95)(5,60,138,68,96)(6,61,139,69,97)(7,62,140,70,98)(8,32,141,71,99)(9,33,142,72,100)(10,34,143,73,101)(11,35,144,74,102)(12,36,145,75,103)(13,37,146,76,104)(14,38,147,77,105)(15,39,148,78,106)(16,40,149,79,107)(17,41,150,80,108)(18,42,151,81,109)(19,43,152,82,110)(20,44,153,83,111)(21,45,154,84,112)(22,46,155,85,113)(23,47,125,86,114)(24,48,126,87,115)(25,49,127,88,116)(26,50,128,89,117)(27,51,129,90,118)(28,52,130,91,119)(29,53,131,92,120)(30,54,132,93,121)(31,55,133,63,122), (1,123)(2,124)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,121)(31,122)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70) );
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)], [(1,56,134,64,123),(2,57,135,65,124),(3,58,136,66,94),(4,59,137,67,95),(5,60,138,68,96),(6,61,139,69,97),(7,62,140,70,98),(8,32,141,71,99),(9,33,142,72,100),(10,34,143,73,101),(11,35,144,74,102),(12,36,145,75,103),(13,37,146,76,104),(14,38,147,77,105),(15,39,148,78,106),(16,40,149,79,107),(17,41,150,80,108),(18,42,151,81,109),(19,43,152,82,110),(20,44,153,83,111),(21,45,154,84,112),(22,46,155,85,113),(23,47,125,86,114),(24,48,126,87,115),(25,49,127,88,116),(26,50,128,89,117),(27,51,129,90,118),(28,52,130,91,119),(29,53,131,92,120),(30,54,132,93,121),(31,55,133,63,122)], [(1,123),(2,124),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,121),(31,122),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70)]])
124 conjugacy classes
| class | 1 | 2 | 5A | 5B | 31A | ··· | 31AD | 62A | ··· | 62AD | 155A | ··· | 155BH |
| order | 1 | 2 | 5 | 5 | 31 | ··· | 31 | 62 | ··· | 62 | 155 | ··· | 155 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C31 | C62 | D5 | D5×C31 |
| kernel | D5×C31 | C155 | D5 | C5 | C31 | C1 |
| # reps | 1 | 1 | 30 | 30 | 2 | 60 |
Matrix representation of D5×C31 ►in GL2(𝔽311) generated by
| 270 | 0 |
| 0 | 270 |
| 310 | 60 |
| 310 | 59 |
| 310 | 0 |
| 310 | 1 |
G:=sub<GL(2,GF(311))| [270,0,0,270],[310,310,60,59],[310,310,0,1] >;
D5×C31 in GAP, Magma, Sage, TeX
D_5\times C_{31} % in TeX
G:=Group("D5xC31"); // GroupNames label
G:=SmallGroup(310,3);
// by ID
G=gap.SmallGroup(310,3);
# by ID
G:=PCGroup([3,-2,-31,-5,2234]);
// Polycyclic
G:=Group<a,b,c|a^31=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export