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G = C6×D33order 396 = 22·32·11

Direct product of C6 and D33

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

Aliases: C6×D33, C661C6, C662S3, C337D6, C325D22, C22⋊(C3×S3), C6⋊(C3×D11), C112(S3×C6), C332(C2×C6), (C3×C66)⋊2C2, C32(C6×D11), (C3×C6)⋊1D11, (C3×C33)⋊7C22, SmallGroup(396,27)

Series: Derived Chief Lower central Upper central

C1C33 — C6×D33
C1C11C33C3×C33C3×D33 — C6×D33
C33 — C6×D33
C1C6

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

33C2
33C2
2C3
33C22
2C6
11S3
11S3
33C6
33C6
3D11
3D11
2C33
11D6
33C2×C6
11C3×S3
11C3×S3
3D22
2C66
3C3×D11
3C3×D11
11S3×C6
3C6×D11

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

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

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

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

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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