direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C5×C31⋊C3, C155⋊C3, C31⋊3C15, SmallGroup(465,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C5×C31⋊C3 |
Generators and relations for C5×C31⋊C3
G = < a,b,c | a5=b31=c3=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C5×C31⋊C3
►On 155 points
Generators in S155
(1 125 94 63 32)(2 126 95 64 33)(3 127 96 65 34)(4 128 97 66 35)(5 129 98 67 36)(6 130 99 68 37)(7 131 100 69 38)(8 132 101 70 39)(9 133 102 71 40)(10 134 103 72 41)(11 135 104 73 42)(12 136 105 74 43)(13 137 106 75 44)(14 138 107 76 45)(15 139 108 77 46)(16 140 109 78 47)(17 141 110 79 48)(18 142 111 80 49)(19 143 112 81 50)(20 144 113 82 51)(21 145 114 83 52)(22 146 115 84 53)(23 147 116 85 54)(24 148 117 86 55)(25 149 118 87 56)(26 150 119 88 57)(27 151 120 89 58)(28 152 121 90 59)(29 153 122 91 60)(30 154 123 92 61)(31 155 124 93 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)
(2 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)(33 57 37)(34 51 42)(35 45 47)(36 39 52)(38 58 62)(40 46 41)(43 59 56)(44 53 61)(48 60 50)(49 54 55)(64 88 68)(65 82 73)(66 76 78)(67 70 83)(69 89 93)(71 77 72)(74 90 87)(75 84 92)(79 91 81)(80 85 86)(95 119 99)(96 113 104)(97 107 109)(98 101 114)(100 120 124)(102 108 103)(105 121 118)(106 115 123)(110 122 112)(111 116 117)(126 150 130)(127 144 135)(128 138 140)(129 132 145)(131 151 155)(133 139 134)(136 152 149)(137 146 154)(141 153 143)(142 147 148)
G:=sub<Sym(155)| (1,125,94,63,32)(2,126,95,64,33)(3,127,96,65,34)(4,128,97,66,35)(5,129,98,67,36)(6,130,99,68,37)(7,131,100,69,38)(8,132,101,70,39)(9,133,102,71,40)(10,134,103,72,41)(11,135,104,73,42)(12,136,105,74,43)(13,137,106,75,44)(14,138,107,76,45)(15,139,108,77,46)(16,140,109,78,47)(17,141,110,79,48)(18,142,111,80,49)(19,143,112,81,50)(20,144,113,82,51)(21,145,114,83,52)(22,146,115,84,53)(23,147,116,85,54)(24,148,117,86,55)(25,149,118,87,56)(26,150,119,88,57)(27,151,120,89,58)(28,152,121,90,59)(29,153,122,91,60)(30,154,123,92,61)(31,155,124,93,62), (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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)(95,119,99)(96,113,104)(97,107,109)(98,101,114)(100,120,124)(102,108,103)(105,121,118)(106,115,123)(110,122,112)(111,116,117)(126,150,130)(127,144,135)(128,138,140)(129,132,145)(131,151,155)(133,139,134)(136,152,149)(137,146,154)(141,153,143)(142,147,148)>;
G:=Group( (1,125,94,63,32)(2,126,95,64,33)(3,127,96,65,34)(4,128,97,66,35)(5,129,98,67,36)(6,130,99,68,37)(7,131,100,69,38)(8,132,101,70,39)(9,133,102,71,40)(10,134,103,72,41)(11,135,104,73,42)(12,136,105,74,43)(13,137,106,75,44)(14,138,107,76,45)(15,139,108,77,46)(16,140,109,78,47)(17,141,110,79,48)(18,142,111,80,49)(19,143,112,81,50)(20,144,113,82,51)(21,145,114,83,52)(22,146,115,84,53)(23,147,116,85,54)(24,148,117,86,55)(25,149,118,87,56)(26,150,119,88,57)(27,151,120,89,58)(28,152,121,90,59)(29,153,122,91,60)(30,154,123,92,61)(31,155,124,93,62), (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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)(95,119,99)(96,113,104)(97,107,109)(98,101,114)(100,120,124)(102,108,103)(105,121,118)(106,115,123)(110,122,112)(111,116,117)(126,150,130)(127,144,135)(128,138,140)(129,132,145)(131,151,155)(133,139,134)(136,152,149)(137,146,154)(141,153,143)(142,147,148) );
G=PermutationGroup([[(1,125,94,63,32),(2,126,95,64,33),(3,127,96,65,34),(4,128,97,66,35),(5,129,98,67,36),(6,130,99,68,37),(7,131,100,69,38),(8,132,101,70,39),(9,133,102,71,40),(10,134,103,72,41),(11,135,104,73,42),(12,136,105,74,43),(13,137,106,75,44),(14,138,107,76,45),(15,139,108,77,46),(16,140,109,78,47),(17,141,110,79,48),(18,142,111,80,49),(19,143,112,81,50),(20,144,113,82,51),(21,145,114,83,52),(22,146,115,84,53),(23,147,116,85,54),(24,148,117,86,55),(25,149,118,87,56),(26,150,119,88,57),(27,151,120,89,58),(28,152,121,90,59),(29,153,122,91,60),(30,154,123,92,61),(31,155,124,93,62)], [(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)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24),(33,57,37),(34,51,42),(35,45,47),(36,39,52),(38,58,62),(40,46,41),(43,59,56),(44,53,61),(48,60,50),(49,54,55),(64,88,68),(65,82,73),(66,76,78),(67,70,83),(69,89,93),(71,77,72),(74,90,87),(75,84,92),(79,91,81),(80,85,86),(95,119,99),(96,113,104),(97,107,109),(98,101,114),(100,120,124),(102,108,103),(105,121,118),(106,115,123),(110,122,112),(111,116,117),(126,150,130),(127,144,135),(128,138,140),(129,132,145),(131,151,155),(133,139,134),(136,152,149),(137,146,154),(141,153,143),(142,147,148)]])
65 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 15A | ··· | 15H | 31A | ··· | 31J | 155A | ··· | 155AN |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 31 | ··· | 31 | 155 | ··· | 155 |
| size | 1 | 31 | 31 | 1 | 1 | 1 | 1 | 31 | ··· | 31 | 3 | ··· | 3 | 3 | ··· | 3 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C31⋊C3 | C5×C31⋊C3 |
| kernel | C5×C31⋊C3 | C155 | C31⋊C3 | C31 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 10 | 40 |
Matrix representation of C5×C31⋊C3 ►in GL3(𝔽1861) generated by
| 739 | 0 | 0 |
| 0 | 739 | 0 |
| 0 | 0 | 739 |
| 1717 | 1637 | 380 |
| 1 | 0 | 374 |
| 0 | 1 | 813 |
| 1520 | 912 | 1050 |
| 479 | 1642 | 1588 |
| 983 | 789 | 560 |
G:=sub<GL(3,GF(1861))| [739,0,0,0,739,0,0,0,739],[1717,1,0,1637,0,1,380,374,813],[1520,479,983,912,1642,789,1050,1588,560] >;
C5×C31⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_{31}\rtimes C_3 % in TeX
G:=Group("C5xC31:C3"); // GroupNames label
G:=SmallGroup(465,3);
// by ID
G=gap.SmallGroup(465,3);
# by ID
G:=PCGroup([3,-3,-5,-31,3377]);
// Polycyclic
G:=Group<a,b,c|a^5=b^31=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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