direct product, abelian, monomial, 5-elementary
Aliases: C5×C30, SmallGroup(150,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C30 |
| C1 — C5×C30 |
| C1 — C5×C30 |
Generators and relations for C5×C30
G = < a,b | a5=b30=1, ab=ba >
Smallest permutation representation of C5×C30
►Regular action on 150 points
Generators in S150
(1 119 63 53 145)(2 120 64 54 146)(3 91 65 55 147)(4 92 66 56 148)(5 93 67 57 149)(6 94 68 58 150)(7 95 69 59 121)(8 96 70 60 122)(9 97 71 31 123)(10 98 72 32 124)(11 99 73 33 125)(12 100 74 34 126)(13 101 75 35 127)(14 102 76 36 128)(15 103 77 37 129)(16 104 78 38 130)(17 105 79 39 131)(18 106 80 40 132)(19 107 81 41 133)(20 108 82 42 134)(21 109 83 43 135)(22 110 84 44 136)(23 111 85 45 137)(24 112 86 46 138)(25 113 87 47 139)(26 114 88 48 140)(27 115 89 49 141)(28 116 90 50 142)(29 117 61 51 143)(30 118 62 52 144)
(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)
G:=sub<Sym(150)| (1,119,63,53,145)(2,120,64,54,146)(3,91,65,55,147)(4,92,66,56,148)(5,93,67,57,149)(6,94,68,58,150)(7,95,69,59,121)(8,96,70,60,122)(9,97,71,31,123)(10,98,72,32,124)(11,99,73,33,125)(12,100,74,34,126)(13,101,75,35,127)(14,102,76,36,128)(15,103,77,37,129)(16,104,78,38,130)(17,105,79,39,131)(18,106,80,40,132)(19,107,81,41,133)(20,108,82,42,134)(21,109,83,43,135)(22,110,84,44,136)(23,111,85,45,137)(24,112,86,46,138)(25,113,87,47,139)(26,114,88,48,140)(27,115,89,49,141)(28,116,90,50,142)(29,117,61,51,143)(30,118,62,52,144), (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)>;
G:=Group( (1,119,63,53,145)(2,120,64,54,146)(3,91,65,55,147)(4,92,66,56,148)(5,93,67,57,149)(6,94,68,58,150)(7,95,69,59,121)(8,96,70,60,122)(9,97,71,31,123)(10,98,72,32,124)(11,99,73,33,125)(12,100,74,34,126)(13,101,75,35,127)(14,102,76,36,128)(15,103,77,37,129)(16,104,78,38,130)(17,105,79,39,131)(18,106,80,40,132)(19,107,81,41,133)(20,108,82,42,134)(21,109,83,43,135)(22,110,84,44,136)(23,111,85,45,137)(24,112,86,46,138)(25,113,87,47,139)(26,114,88,48,140)(27,115,89,49,141)(28,116,90,50,142)(29,117,61,51,143)(30,118,62,52,144), (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) );
G=PermutationGroup([[(1,119,63,53,145),(2,120,64,54,146),(3,91,65,55,147),(4,92,66,56,148),(5,93,67,57,149),(6,94,68,58,150),(7,95,69,59,121),(8,96,70,60,122),(9,97,71,31,123),(10,98,72,32,124),(11,99,73,33,125),(12,100,74,34,126),(13,101,75,35,127),(14,102,76,36,128),(15,103,77,37,129),(16,104,78,38,130),(17,105,79,39,131),(18,106,80,40,132),(19,107,81,41,133),(20,108,82,42,134),(21,109,83,43,135),(22,110,84,44,136),(23,111,85,45,137),(24,112,86,46,138),(25,113,87,47,139),(26,114,88,48,140),(27,115,89,49,141),(28,116,90,50,142),(29,117,61,51,143),(30,118,62,52,144)], [(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)]])
C5×C30 is a maximal subgroup of
C30.D5
150 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5X | 6A | 6B | 10A | ··· | 10X | 15A | ··· | 15AV | 30A | ··· | 30AV |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C5×C30 | C5×C15 | C5×C10 | C30 | C52 | C15 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 2 | 24 | 48 | 48 |
Matrix representation of C5×C30 ►in GL2(𝔽31) generated by
| 4 | 0 |
| 0 | 8 |
| 13 | 0 |
| 0 | 4 |
G:=sub<GL(2,GF(31))| [4,0,0,8],[13,0,0,4] >;
C5×C30 in GAP, Magma, Sage, TeX
C_5\times C_{30} % in TeX
G:=Group("C5xC30"); // GroupNames label
G:=SmallGroup(150,13);
// by ID
G=gap.SmallGroup(150,13);
# by ID
G:=PCGroup([4,-2,-3,-5,-5]);
// Polycyclic
G:=Group<a,b|a^5=b^30=1,a*b=b*a>;
// generators/relations
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