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G = C6×C30order 180 = 22·32·5

Abelian group of type [6,30]

direct product, abelian, monomial

Aliases: C6×C30, SmallGroup(180,37)

Series: Derived Chief Lower central Upper central

C1 — C6×C30
C1C5C15C3×C15C3×C30 — C6×C30
C1 — C6×C30
C1 — C6×C30

Generators and relations for C6×C30
 G = < a,b | a6=b30=1, ab=ba >

Subgroups: 60, all normal (8 characteristic)
C1, C2 [×3], C3 [×4], C22, C5, C6 [×12], C32, C10 [×3], C2×C6 [×4], C15 [×4], C3×C6 [×3], C2×C10, C30 [×12], C62, C3×C15, C2×C30 [×4], C3×C30 [×3], C6×C30
Quotients: C1, C2 [×3], C3 [×4], C22, C5, C6 [×12], C32, C10 [×3], C2×C6 [×4], C15 [×4], C3×C6 [×3], C2×C10, C30 [×12], C62, C3×C15, C2×C30 [×4], C3×C30 [×3], C6×C30

Smallest permutation representation of C6×C30
Regular action on 180 points
Generators in S180
(1 80 91 43 150 169)(2 81 92 44 121 170)(3 82 93 45 122 171)(4 83 94 46 123 172)(5 84 95 47 124 173)(6 85 96 48 125 174)(7 86 97 49 126 175)(8 87 98 50 127 176)(9 88 99 51 128 177)(10 89 100 52 129 178)(11 90 101 53 130 179)(12 61 102 54 131 180)(13 62 103 55 132 151)(14 63 104 56 133 152)(15 64 105 57 134 153)(16 65 106 58 135 154)(17 66 107 59 136 155)(18 67 108 60 137 156)(19 68 109 31 138 157)(20 69 110 32 139 158)(21 70 111 33 140 159)(22 71 112 34 141 160)(23 72 113 35 142 161)(24 73 114 36 143 162)(25 74 115 37 144 163)(26 75 116 38 145 164)(27 76 117 39 146 165)(28 77 118 40 147 166)(29 78 119 41 148 167)(30 79 120 42 149 168)
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)

G:=sub<Sym(180)| (1,80,91,43,150,169)(2,81,92,44,121,170)(3,82,93,45,122,171)(4,83,94,46,123,172)(5,84,95,47,124,173)(6,85,96,48,125,174)(7,86,97,49,126,175)(8,87,98,50,127,176)(9,88,99,51,128,177)(10,89,100,52,129,178)(11,90,101,53,130,179)(12,61,102,54,131,180)(13,62,103,55,132,151)(14,63,104,56,133,152)(15,64,105,57,134,153)(16,65,106,58,135,154)(17,66,107,59,136,155)(18,67,108,60,137,156)(19,68,109,31,138,157)(20,69,110,32,139,158)(21,70,111,33,140,159)(22,71,112,34,141,160)(23,72,113,35,142,161)(24,73,114,36,143,162)(25,74,115,37,144,163)(26,75,116,38,145,164)(27,76,117,39,146,165)(28,77,118,40,147,166)(29,78,119,41,148,167)(30,79,120,42,149,168), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)>;

G:=Group( (1,80,91,43,150,169)(2,81,92,44,121,170)(3,82,93,45,122,171)(4,83,94,46,123,172)(5,84,95,47,124,173)(6,85,96,48,125,174)(7,86,97,49,126,175)(8,87,98,50,127,176)(9,88,99,51,128,177)(10,89,100,52,129,178)(11,90,101,53,130,179)(12,61,102,54,131,180)(13,62,103,55,132,151)(14,63,104,56,133,152)(15,64,105,57,134,153)(16,65,106,58,135,154)(17,66,107,59,136,155)(18,67,108,60,137,156)(19,68,109,31,138,157)(20,69,110,32,139,158)(21,70,111,33,140,159)(22,71,112,34,141,160)(23,72,113,35,142,161)(24,73,114,36,143,162)(25,74,115,37,144,163)(26,75,116,38,145,164)(27,76,117,39,146,165)(28,77,118,40,147,166)(29,78,119,41,148,167)(30,79,120,42,149,168), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180) );

G=PermutationGroup([(1,80,91,43,150,169),(2,81,92,44,121,170),(3,82,93,45,122,171),(4,83,94,46,123,172),(5,84,95,47,124,173),(6,85,96,48,125,174),(7,86,97,49,126,175),(8,87,98,50,127,176),(9,88,99,51,128,177),(10,89,100,52,129,178),(11,90,101,53,130,179),(12,61,102,54,131,180),(13,62,103,55,132,151),(14,63,104,56,133,152),(15,64,105,57,134,153),(16,65,106,58,135,154),(17,66,107,59,136,155),(18,67,108,60,137,156),(19,68,109,31,138,157),(20,69,110,32,139,158),(21,70,111,33,140,159),(22,71,112,34,141,160),(23,72,113,35,142,161),(24,73,114,36,143,162),(25,74,115,37,144,163),(26,75,116,38,145,164),(27,76,117,39,146,165),(28,77,118,40,147,166),(29,78,119,41,148,167),(30,79,120,42,149,168)], [(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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)])

C6×C30 is a maximal subgroup of   C62⋊D5

180 conjugacy classes

class 1 2A2B2C3A···3H5A5B5C5D6A···6X10A···10L15A···15AF30A···30CR
order12223···355556···610···1015···1530···30
size11111···111111···11···11···11···1

180 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC6×C30C3×C30C2×C30C62C30C3×C6C2×C6C6
# reps138424123296

Matrix representation of C6×C30 in GL2(𝔽31) generated by

260
05
,
50
015
G:=sub<GL(2,GF(31))| [26,0,0,5],[5,0,0,15] >;

C6×C30 in GAP, Magma, Sage, TeX

C_6\times C_{30}
% in TeX

G:=Group("C6xC30");
// GroupNames label

G:=SmallGroup(180,37);
// by ID

G=gap.SmallGroup(180,37);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5]);
// Polycyclic

G:=Group<a,b|a^6=b^30=1,a*b=b*a>;
// generators/relations

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