Copied to
clipboard

G = C5×D33order 330 = 2·3·5·11

Direct product of C5 and D33

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

Aliases: C5×D33, C552S3, C1653C2, C335C10, C153D11, C3⋊(C5×D11), C113(C5×S3), SmallGroup(330,9)

Series: Derived Chief Lower central Upper central

C1C33 — C5×D33
C1C11C33C165 — C5×D33
C33 — C5×D33
C1C5

Generators and relations for C5×D33
 G = < a,b,c | a5=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

33C2
11S3
33C10
3D11
11C5×S3
3C5×D11

Smallest permutation representation of C5×D33
On 165 points
Generators in S165
(1 145 100 91 50)(2 146 101 92 51)(3 147 102 93 52)(4 148 103 94 53)(5 149 104 95 54)(6 150 105 96 55)(7 151 106 97 56)(8 152 107 98 57)(9 153 108 99 58)(10 154 109 67 59)(11 155 110 68 60)(12 156 111 69 61)(13 157 112 70 62)(14 158 113 71 63)(15 159 114 72 64)(16 160 115 73 65)(17 161 116 74 66)(18 162 117 75 34)(19 163 118 76 35)(20 164 119 77 36)(21 165 120 78 37)(22 133 121 79 38)(23 134 122 80 39)(24 135 123 81 40)(25 136 124 82 41)(26 137 125 83 42)(27 138 126 84 43)(28 139 127 85 44)(29 140 128 86 45)(30 141 129 87 46)(31 142 130 88 47)(32 143 131 89 48)(33 144 132 90 49)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(100 132)(101 131)(102 130)(103 129)(104 128)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 117)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(157 165)(158 164)(159 163)(160 162)

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

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

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

90 conjugacy classes

class 1  2  3 5A5B5C5D10A10B10C10D11A···11E15A15B15C15D33A···33J55A···55T165A···165AN
order12355551010101011···111515151533···3355···55165···165
size13321111333333332···222222···22···22···2

90 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10S3D11C5×S3D33C5×D11C5×D33
kernelC5×D33C165D33C33C55C15C11C5C3C1
# reps1144154102040

Matrix representation of C5×D33 in GL2(𝔽331) generated by

3230
0323
,
188250
81113
,
201232
271130
G:=sub<GL(2,GF(331))| [323,0,0,323],[188,81,250,113],[201,271,232,130] >;

C5×D33 in GAP, Magma, Sage, TeX

C_5\times D_{33}
% in TeX

G:=Group("C5xD33");
// GroupNames label

G:=SmallGroup(330,9);
// by ID

G=gap.SmallGroup(330,9);
# by ID

G:=PCGroup([4,-2,-5,-3,-11,242,4803]);
// Polycyclic

G:=Group<a,b,c|a^5=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D33 in TeX

׿
×
𝔽