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G = C3×D55order 330 = 2·3·5·11

Direct product of C3 and D55

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

Aliases: C3×D55, C551C6, C332D5, C1652C2, C152D11, C11⋊(C3×D5), C5⋊(C3×D11), SmallGroup(330,7)

Series: Derived Chief Lower central Upper central

C1C55 — C3×D55
C1C11C55C165 — C3×D55
C55 — C3×D55
C1C3

Generators and relations for C3×D55
 G = < a,b,c | a3=b55=c2=1, ab=ba, ac=ca, cbc=b-1 >

55C2
55C6
11D5
5D11
11C3×D5
5C3×D11

Smallest permutation representation of C3×D55
On 165 points
Generators in S165
(1 141 68)(2 142 69)(3 143 70)(4 144 71)(5 145 72)(6 146 73)(7 147 74)(8 148 75)(9 149 76)(10 150 77)(11 151 78)(12 152 79)(13 153 80)(14 154 81)(15 155 82)(16 156 83)(17 157 84)(18 158 85)(19 159 86)(20 160 87)(21 161 88)(22 162 89)(23 163 90)(24 164 91)(25 165 92)(26 111 93)(27 112 94)(28 113 95)(29 114 96)(30 115 97)(31 116 98)(32 117 99)(33 118 100)(34 119 101)(35 120 102)(36 121 103)(37 122 104)(38 123 105)(39 124 106)(40 125 107)(41 126 108)(42 127 109)(43 128 110)(44 129 56)(45 130 57)(46 131 58)(47 132 59)(48 133 60)(49 134 61)(50 135 62)(51 136 63)(52 137 64)(53 138 65)(54 139 66)(55 140 67)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(111 115)(112 114)(116 165)(117 164)(118 163)(119 162)(120 161)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)

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

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

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

87 conjugacy classes

class 1  2 3A3B5A5B6A6B11A···11E15A15B15C15D33A···33J55A···55T165A···165AN
order1233556611···111515151533···3355···55165···165
size155112255552···222222···22···22···2

87 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6D5D11C3×D5C3×D11D55C3×D55
kernelC3×D55C165D55C55C33C15C11C5C3C1
# reps1122254102040

Matrix representation of C3×D55 in GL2(𝔽331) generated by

310
031
,
205325
6129
,
205325
53126
G:=sub<GL(2,GF(331))| [31,0,0,31],[205,6,325,129],[205,53,325,126] >;

C3×D55 in GAP, Magma, Sage, TeX

C_3\times D_{55}
% in TeX

G:=Group("C3xD55");
// GroupNames label

G:=SmallGroup(330,7);
// by ID

G=gap.SmallGroup(330,7);
# by ID

G:=PCGroup([4,-2,-3,-5,-11,290,4803]);
// Polycyclic

G:=Group<a,b,c|a^3=b^55=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D55 in TeX

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