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G = S3×C66order 396 = 22·32·11

Direct product of C66 and S3

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

Aliases: S3×C66, C6⋊C66, C663C6, C3⋊(C2×C66), (C3×C6)⋊1C22, (C3×C66)⋊4C2, C334(C2×C6), (C3×C33)⋊9C22, C322(C2×C22), SmallGroup(396,26)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C66
C1C3C33C3×C33S3×C33 — S3×C66
C3 — S3×C66
C1C66

Generators and relations for S3×C66
 G = < a,b,c | a66=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C22
2C6
3C6
3C6
3C22
3C22
2C33
3C2×C6
3C2×C22
2C66
3C66
3C66
3C2×C66

Smallest permutation representation of S3×C66
On 132 points
Generators in S132
(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 45 23)(2 46 24)(3 47 25)(4 48 26)(5 49 27)(6 50 28)(7 51 29)(8 52 30)(9 53 31)(10 54 32)(11 55 33)(12 56 34)(13 57 35)(14 58 36)(15 59 37)(16 60 38)(17 61 39)(18 62 40)(19 63 41)(20 64 42)(21 65 43)(22 66 44)(67 89 111)(68 90 112)(69 91 113)(70 92 114)(71 93 115)(72 94 116)(73 95 117)(74 96 118)(75 97 119)(76 98 120)(77 99 121)(78 100 122)(79 101 123)(80 102 124)(81 103 125)(82 104 126)(83 105 127)(84 106 128)(85 107 129)(86 108 130)(87 109 131)(88 110 132)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)

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

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

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

198 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I11A···11J22A···22J22K···22AD33A···33T33U···33AX66A···66T66U···66AX66AY···66CL
order12223333366666666611···1122···2222···2233···3333···3366···6666···6666···66
size1133112221122233331···11···13···31···12···21···12···23···3

198 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C6C6C11C22C22C33C66C66S3D6C3×S3S3×C6S3×C11S3×C22S3×C33S3×C66
kernelS3×C66S3×C33C3×C66S3×C22S3×C11C66S3×C6C3×S3C3×C6D6S3C6C66C33C22C11C6C3C2C1
# reps121242102010204020112210102020

Matrix representation of S3×C66 in GL2(𝔽67) generated by

610
061
,
290
3037
,
136
066
G:=sub<GL(2,GF(67))| [61,0,0,61],[29,30,0,37],[1,0,36,66] >;

S3×C66 in GAP, Magma, Sage, TeX

S_3\times C_{66}
% in TeX

G:=Group("S3xC66");
// GroupNames label

G:=SmallGroup(396,26);
// by ID

G=gap.SmallGroup(396,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-3,6604]);
// Polycyclic

G:=Group<a,b,c|a^66=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C66 in TeX

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