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G = C15×D11order 330 = 2·3·5·11

Direct product of C15 and D11

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C15×D11, C552C6, C1655C2, C113C30, C336C10, SmallGroup(330,5)

Series: Derived Chief Lower central Upper central

C1C11 — C15×D11
C1C11C55C165 — C15×D11
C11 — C15×D11
C1C15

Generators and relations for C15×D11
 G = < a,b,c | a15=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C6
11C10
11C30

Smallest permutation representation of C15×D11
On 165 points
Generators in S165
(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)
(1 115 67 150 123 98 158 78 58 38 27)(2 116 68 136 124 99 159 79 59 39 28)(3 117 69 137 125 100 160 80 60 40 29)(4 118 70 138 126 101 161 81 46 41 30)(5 119 71 139 127 102 162 82 47 42 16)(6 120 72 140 128 103 163 83 48 43 17)(7 106 73 141 129 104 164 84 49 44 18)(8 107 74 142 130 105 165 85 50 45 19)(9 108 75 143 131 91 151 86 51 31 20)(10 109 61 144 132 92 152 87 52 32 21)(11 110 62 145 133 93 153 88 53 33 22)(12 111 63 146 134 94 154 89 54 34 23)(13 112 64 147 135 95 155 90 55 35 24)(14 113 65 148 121 96 156 76 56 36 25)(15 114 66 149 122 97 157 77 57 37 26)
(1 27)(2 28)(3 29)(4 30)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 106)(45 107)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(76 148)(77 149)(78 150)(79 136)(80 137)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 145)(89 146)(90 147)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 151)(132 152)(133 153)(134 154)(135 155)

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

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

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

105 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B10A10B10C10D11A···11E15A···15H30A···30H33A···33J55A···55T165A···165AN
order12335555661010101011···1115···1530···3033···3355···55165···165
size1111111111111111111112···21···111···112···22···22···2

105 irreducible representations

dim111111112222
type+++
imageC1C2C3C5C6C10C15C30D11C3×D11C5×D11C15×D11
kernelC15×D11C165C5×D11C3×D11C55C33D11C11C15C5C3C1
# reps112424885102040

Matrix representation of C15×D11 in GL3(𝔽331) generated by

32300
02990
00299
,
100
001
0330123
,
33000
001
010
G:=sub<GL(3,GF(331))| [323,0,0,0,299,0,0,0,299],[1,0,0,0,0,330,0,1,123],[330,0,0,0,0,1,0,1,0] >;

C15×D11 in GAP, Magma, Sage, TeX

C_{15}\times D_{11}
% in TeX

G:=Group("C15xD11");
// GroupNames label

G:=SmallGroup(330,5);
// by ID

G=gap.SmallGroup(330,5);
# by ID

G:=PCGroup([4,-2,-3,-5,-11,4803]);
// Polycyclic

G:=Group<a,b,c|a^15=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×D11 in TeX

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