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G = C6×D33  order 396 = 22·32·11

Direct product of C6 and D33

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C6×D33, C66⋊1C6, C66⋊2S3, C33⋊7D6, C32⋊5D22, C22⋊(C3×S3), C6⋊(C3×D11), C11⋊2(S3×C6), C33⋊2(C2×C6), (C3×C66)⋊2C2, C3⋊2(C6×D11), (C3×C6)⋊1D11, (C3×C33)⋊7C22, SmallGroup(396,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C6×D33
Chief series
C1 — C11 — C33 — C3×C33 — C3×D33 — C6×D33
Lower central
C33 — C6×D33
Upper central
C1 — C6

Generators and relations for C6×D33
 G = < a,b,c | a6=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
C2
33C2
33C2
C3
C3
2C3
C11
33C22
C6
C6
2C6
11S3
11S3
33C6
33C6
C32
C22
3D11
3D11
C33
C33
2C33
11D6
33C2×C6
C3×C6
11C3×S3
11C3×S3
3D22
C66
C66
D33
D33
2C66
3C3×D11
3C3×D11
C3×C33
11S3×C6
D66
3C6×D11
C3×D33
C3×D33
C3×C66
C6×D33

Smallest permutation representation of C6×D33
►On 132 points
Generators in S132
(1 35 12 46 23 57)(2 36 13 47 24 58)(3 37 14 48 25 59)(4 38 15 49 26 60)(5 39 16 50 27 61)(6 40 17 51 28 62)(7 41 18 52 29 63)(8 42 19 53 30 64)(9 43 20 54 31 65)(10 44 21 55 32 66)(11 45 22 56 33 34)(67 107 89 129 78 118)(68 108 90 130 79 119)(69 109 91 131 80 120)(70 110 92 132 81 121)(71 111 93 100 82 122)(72 112 94 101 83 123)(73 113 95 102 84 124)(74 114 96 103 85 125)(75 115 97 104 86 126)(76 116 98 105 87 127)(77 117 99 106 88 128)

(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)

(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 126)(20 125)(21 124)(22 123)(23 122)(24 121)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 99)(63 98)(64 97)(65 96)(66 95)
 
G:=sub<Sym(132)| (1,35,12,46,23,57)(2,36,13,47,24,58)(3,37,14,48,25,59)(4,38,15,49,26,60)(5,39,16,50,27,61)(6,40,17,51,28,62)(7,41,18,52,29,63)(8,42,19,53,30,64)(9,43,20,54,31,65)(10,44,21,55,32,66)(11,45,22,56,33,34)(67,107,89,129,78,118)(68,108,90,130,79,119)(69,109,91,131,80,120)(70,110,92,132,81,121)(71,111,93,100,82,122)(72,112,94,101,83,123)(73,113,95,102,84,124)(74,114,96,103,85,125)(75,115,97,104,86,126)(76,116,98,105,87,127)(77,117,99,106,88,128), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,99)(63,98)(64,97)(65,96)(66,95)>;
 
G:=Group( (1,35,12,46,23,57)(2,36,13,47,24,58)(3,37,14,48,25,59)(4,38,15,49,26,60)(5,39,16,50,27,61)(6,40,17,51,28,62)(7,41,18,52,29,63)(8,42,19,53,30,64)(9,43,20,54,31,65)(10,44,21,55,32,66)(11,45,22,56,33,34)(67,107,89,129,78,118)(68,108,90,130,79,119)(69,109,91,131,80,120)(70,110,92,132,81,121)(71,111,93,100,82,122)(72,112,94,101,83,123)(73,113,95,102,84,124)(74,114,96,103,85,125)(75,115,97,104,86,126)(76,116,98,105,87,127)(77,117,99,106,88,128), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,99)(63,98)(64,97)(65,96)(66,95) );
 
G=PermutationGroup([[(1,35,12,46,23,57),(2,36,13,47,24,58),(3,37,14,48,25,59),(4,38,15,49,26,60),(5,39,16,50,27,61),(6,40,17,51,28,62),(7,41,18,52,29,63),(8,42,19,53,30,64),(9,43,20,54,31,65),(10,44,21,55,32,66),(11,45,22,56,33,34),(67,107,89,129,78,118),(68,108,90,130,79,119),(69,109,91,131,80,120),(70,110,92,132,81,121),(71,111,93,100,82,122),(72,112,94,101,83,123),(73,113,95,102,84,124),(74,114,96,103,85,125),(75,115,97,104,86,126),(76,116,98,105,87,127),(77,117,99,106,88,128)], [(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)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,100),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,126),(20,125),(21,124),(22,123),(23,122),(24,121),(25,120),(26,119),(27,118),(28,117),(29,116),(30,115),(31,114),(32,113),(33,112),(34,94),(35,93),(36,92),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,84),(45,83),(46,82),(47,81),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,99),(63,98),(64,97),(65,96),(66,95)]])
 

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I11A···11E22A···22E33A···33AN66A···66AN
order12223333366666666611···1122···2233···3366···66
size1133331122211222333333332···22···22···22···2

108 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D6C3×S3D11S3×C6D22C3×D11D33C6×D11D66C3×D33C6×D33
kernelC6×D33C3×D33C3×C66D66D33C66C66C33C22C3×C6C11C32C6C6C3C3C2C1
# reps121242112525101010102020

Matrix representation of C6×D33 ►in GL2(𝔽67) generated by

300
030
,
170
04
,
063
500
G:=sub<GL(2,GF(67))| [30,0,0,30],[17,0,0,4],[0,50,63,0] >;
 

C6×D33 in GAP, Magma, Sage, TeX 

C_6\times D_{33}
 
% in TeX
 
G:=Group("C6xD33");
 
// GroupNames label
 
G:=SmallGroup(396,27);
 
// by ID
 
G=gap.SmallGroup(396,27);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-11,483,9004]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C6×D33 in TeX

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