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G = C11×D15order 330 = 2·3·5·11

Direct product of C11 and D15

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

Aliases: C11×D15, C553S3, C333D5, C1654C2, C151C22, C5⋊(S3×C11), C3⋊(D5×C11), SmallGroup(330,10)

Series: Derived Chief Lower central Upper central

C1C15 — C11×D15
C1C5C15C165 — C11×D15
C15 — C11×D15
C1C11

Generators and relations for C11×D15
 G = < a,b,c | a11=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
5S3
3D5
15C22
5S3×C11
3D5×C11

Smallest permutation representation of C11×D15
On 165 points
Generators in S165
(1 159 144 133 107 91 76 62 58 35 28)(2 160 145 134 108 92 77 63 59 36 29)(3 161 146 135 109 93 78 64 60 37 30)(4 162 147 121 110 94 79 65 46 38 16)(5 163 148 122 111 95 80 66 47 39 17)(6 164 149 123 112 96 81 67 48 40 18)(7 165 150 124 113 97 82 68 49 41 19)(8 151 136 125 114 98 83 69 50 42 20)(9 152 137 126 115 99 84 70 51 43 21)(10 153 138 127 116 100 85 71 52 44 22)(11 154 139 128 117 101 86 72 53 45 23)(12 155 140 129 118 102 87 73 54 31 24)(13 156 141 130 119 103 88 74 55 32 25)(14 157 142 131 120 104 89 75 56 33 26)(15 158 143 132 106 105 90 61 57 34 27)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 107)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(121 129)(122 128)(123 127)(124 126)(130 135)(131 134)(132 133)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(152 165)(153 164)(154 163)(155 162)(156 161)(157 160)(158 159)

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

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

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

99 conjugacy classes

class 1  2  3 5A5B11A···11J15A15B15C15D22A···22J33A···33J55A···55T165A···165AN
order1235511···111515151522···2233···3355···55165···165
size1152221···1222215···152···22···22···2

99 irreducible representations

dim1111222222
type+++++
imageC1C2C11C22S3D5D15S3×C11D5×C11C11×D15
kernelC11×D15C165D15C15C55C33C11C5C3C1
# reps111010124102040

Matrix representation of C11×D15 in GL2(𝔽331) generated by

1800
0180
,
27454
252220
,
3300
11
G:=sub<GL(2,GF(331))| [180,0,0,180],[274,252,54,220],[330,1,0,1] >;

C11×D15 in GAP, Magma, Sage, TeX

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

G:=Group("C11xD15");
// GroupNames label

G:=SmallGroup(330,10);
// by ID

G=gap.SmallGroup(330,10);
# by ID

G:=PCGroup([4,-2,-11,-3,-5,530,4227]);
// Polycyclic

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

Export

Subgroup lattice of C11×D15 in TeX

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