direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D31, C31⋊3C10, C155⋊2C2, SmallGroup(310,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C5×D31 |
Generators and relations for C5×D31
G = < a,b,c | a5=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D31
►On 155 points
Generators in S155
(1 134 104 83 47)(2 135 105 84 48)(3 136 106 85 49)(4 137 107 86 50)(5 138 108 87 51)(6 139 109 88 52)(7 140 110 89 53)(8 141 111 90 54)(9 142 112 91 55)(10 143 113 92 56)(11 144 114 93 57)(12 145 115 63 58)(13 146 116 64 59)(14 147 117 65 60)(15 148 118 66 61)(16 149 119 67 62)(17 150 120 68 32)(18 151 121 69 33)(19 152 122 70 34)(20 153 123 71 35)(21 154 124 72 36)(22 155 94 73 37)(23 125 95 74 38)(24 126 96 75 39)(25 127 97 76 40)(26 128 98 77 41)(27 129 99 78 42)(28 130 100 79 43)(29 131 101 80 44)(30 132 102 81 45)(31 133 103 82 46)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(63 71)(64 70)(65 69)(66 68)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(114 124)(115 123)(116 122)(117 121)(118 120)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)
G:=sub<Sym(155)| (1,134,104,83,47)(2,135,105,84,48)(3,136,106,85,49)(4,137,107,86,50)(5,138,108,87,51)(6,139,109,88,52)(7,140,110,89,53)(8,141,111,90,54)(9,142,112,91,55)(10,143,113,92,56)(11,144,114,93,57)(12,145,115,63,58)(13,146,116,64,59)(14,147,117,65,60)(15,148,118,66,61)(16,149,119,67,62)(17,150,120,68,32)(18,151,121,69,33)(19,152,122,70,34)(20,153,123,71,35)(21,154,124,72,36)(22,155,94,73,37)(23,125,95,74,38)(24,126,96,75,39)(25,127,97,76,40)(26,128,98,77,41)(27,129,99,78,42)(28,130,100,79,43)(29,131,101,80,44)(30,132,102,81,45)(31,133,103,82,46), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(63,71)(64,70)(65,69)(66,68)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(114,124)(115,123)(116,122)(117,121)(118,120)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)>;
G:=Group( (1,134,104,83,47)(2,135,105,84,48)(3,136,106,85,49)(4,137,107,86,50)(5,138,108,87,51)(6,139,109,88,52)(7,140,110,89,53)(8,141,111,90,54)(9,142,112,91,55)(10,143,113,92,56)(11,144,114,93,57)(12,145,115,63,58)(13,146,116,64,59)(14,147,117,65,60)(15,148,118,66,61)(16,149,119,67,62)(17,150,120,68,32)(18,151,121,69,33)(19,152,122,70,34)(20,153,123,71,35)(21,154,124,72,36)(22,155,94,73,37)(23,125,95,74,38)(24,126,96,75,39)(25,127,97,76,40)(26,128,98,77,41)(27,129,99,78,42)(28,130,100,79,43)(29,131,101,80,44)(30,132,102,81,45)(31,133,103,82,46), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(63,71)(64,70)(65,69)(66,68)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(114,124)(115,123)(116,122)(117,121)(118,120)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150) );
G=PermutationGroup([[(1,134,104,83,47),(2,135,105,84,48),(3,136,106,85,49),(4,137,107,86,50),(5,138,108,87,51),(6,139,109,88,52),(7,140,110,89,53),(8,141,111,90,54),(9,142,112,91,55),(10,143,113,92,56),(11,144,114,93,57),(12,145,115,63,58),(13,146,116,64,59),(14,147,117,65,60),(15,148,118,66,61),(16,149,119,67,62),(17,150,120,68,32),(18,151,121,69,33),(19,152,122,70,34),(20,153,123,71,35),(21,154,124,72,36),(22,155,94,73,37),(23,125,95,74,38),(24,126,96,75,39),(25,127,97,76,40),(26,128,98,77,41),(27,129,99,78,42),(28,130,100,79,43),(29,131,101,80,44),(30,132,102,81,45),(31,133,103,82,46)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(63,71),(64,70),(65,69),(66,68),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(114,124),(115,123),(116,122),(117,121),(118,120),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150)]])
85 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 31A | ··· | 31O | 155A | ··· | 155BH |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 31 | ··· | 31 | 155 | ··· | 155 |
| size | 1 | 31 | 1 | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D31 | C5×D31 |
| kernel | C5×D31 | C155 | D31 | C31 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 15 | 60 |
Matrix representation of C5×D31 ►in GL2(𝔽311) generated by
| 6 | 0 |
| 0 | 6 |
| 146 | 134 |
| 310 | 14 |
| 7 | 55 |
| 214 | 304 |
G:=sub<GL(2,GF(311))| [6,0,0,6],[146,310,134,14],[7,214,55,304] >;
C5×D31 in GAP, Magma, Sage, TeX
C_5\times D_{31} % in TeX
G:=Group("C5xD31"); // GroupNames label
G:=SmallGroup(310,4);
// by ID
G=gap.SmallGroup(310,4);
# by ID
G:=PCGroup([3,-2,-5,-31,2702]);
// Polycyclic
G:=Group<a,b,c|a^5=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export