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G = C5×C30order 150 = 2·3·52

Abelian group of type [5,30]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C30, SmallGroup(150,13)

Series: Derived Chief Lower central Upper central

C1 — C5×C30
C1C5C52C5×C15 — 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 91 68 42 136)(2 92 69 43 137)(3 93 70 44 138)(4 94 71 45 139)(5 95 72 46 140)(6 96 73 47 141)(7 97 74 48 142)(8 98 75 49 143)(9 99 76 50 144)(10 100 77 51 145)(11 101 78 52 146)(12 102 79 53 147)(13 103 80 54 148)(14 104 81 55 149)(15 105 82 56 150)(16 106 83 57 121)(17 107 84 58 122)(18 108 85 59 123)(19 109 86 60 124)(20 110 87 31 125)(21 111 88 32 126)(22 112 89 33 127)(23 113 90 34 128)(24 114 61 35 129)(25 115 62 36 130)(26 116 63 37 131)(27 117 64 38 132)(28 118 65 39 133)(29 119 66 40 134)(30 120 67 41 135)
(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)| (1,91,68,42,136)(2,92,69,43,137)(3,93,70,44,138)(4,94,71,45,139)(5,95,72,46,140)(6,96,73,47,141)(7,97,74,48,142)(8,98,75,49,143)(9,99,76,50,144)(10,100,77,51,145)(11,101,78,52,146)(12,102,79,53,147)(13,103,80,54,148)(14,104,81,55,149)(15,105,82,56,150)(16,106,83,57,121)(17,107,84,58,122)(18,108,85,59,123)(19,109,86,60,124)(20,110,87,31,125)(21,111,88,32,126)(22,112,89,33,127)(23,113,90,34,128)(24,114,61,35,129)(25,115,62,36,130)(26,116,63,37,131)(27,117,64,38,132)(28,118,65,39,133)(29,119,66,40,134)(30,120,67,41,135), (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( (1,91,68,42,136)(2,92,69,43,137)(3,93,70,44,138)(4,94,71,45,139)(5,95,72,46,140)(6,96,73,47,141)(7,97,74,48,142)(8,98,75,49,143)(9,99,76,50,144)(10,100,77,51,145)(11,101,78,52,146)(12,102,79,53,147)(13,103,80,54,148)(14,104,81,55,149)(15,105,82,56,150)(16,106,83,57,121)(17,107,84,58,122)(18,108,85,59,123)(19,109,86,60,124)(20,110,87,31,125)(21,111,88,32,126)(22,112,89,33,127)(23,113,90,34,128)(24,114,61,35,129)(25,115,62,36,130)(26,116,63,37,131)(27,117,64,38,132)(28,118,65,39,133)(29,119,66,40,134)(30,120,67,41,135), (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([(1,91,68,42,136),(2,92,69,43,137),(3,93,70,44,138),(4,94,71,45,139),(5,95,72,46,140),(6,96,73,47,141),(7,97,74,48,142),(8,98,75,49,143),(9,99,76,50,144),(10,100,77,51,145),(11,101,78,52,146),(12,102,79,53,147),(13,103,80,54,148),(14,104,81,55,149),(15,105,82,56,150),(16,106,83,57,121),(17,107,84,58,122),(18,108,85,59,123),(19,109,86,60,124),(20,110,87,31,125),(21,111,88,32,126),(22,112,89,33,127),(23,113,90,34,128),(24,114,61,35,129),(25,115,62,36,130),(26,116,63,37,131),(27,117,64,38,132),(28,118,65,39,133),(29,119,66,40,134),(30,120,67,41,135)], [(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 3A3B5A···5X6A6B10A···10X15A···15AV30A···30AV
order12335···56610···1015···1530···30
size11111···1111···11···11···1

150 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC5×C30C5×C15C5×C10C30C52C15C10C5
# reps112242244848

Matrix representation of C5×C30 in GL2(𝔽31) generated by

40
08
,
130
04
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

Export

Subgroup lattice of C5×C30 in TeX

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