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G = S3×C44order 264 = 23·3·11

Direct product of C44 and S3

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

Aliases: S3×C44, D6.C22, C1326C2, C122C22, C22.14D6, Dic32C22, C66.19C22, C31(C2×C44), C336(C2×C4), C2.1(S3×C22), C6.2(C2×C22), (S3×C22).2C2, (C11×Dic3)⋊5C2, SmallGroup(264,19)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C44
C1C3C6C66S3×C22 — S3×C44
C3 — S3×C44
C1C44

Generators and relations for S3×C44
 G = < a,b,c | a44=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C22
3C22
3C2×C4
3C44
3C2×C22
3C2×C44

Smallest permutation representation of S3×C44
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 80 99)(2 81 100)(3 82 101)(4 83 102)(5 84 103)(6 85 104)(7 86 105)(8 87 106)(9 88 107)(10 45 108)(11 46 109)(12 47 110)(13 48 111)(14 49 112)(15 50 113)(16 51 114)(17 52 115)(18 53 116)(19 54 117)(20 55 118)(21 56 119)(22 57 120)(23 58 121)(24 59 122)(25 60 123)(26 61 124)(27 62 125)(28 63 126)(29 64 127)(30 65 128)(31 66 129)(32 67 130)(33 68 131)(34 69 132)(35 70 89)(36 71 90)(37 72 91)(38 73 92)(39 74 93)(40 75 94)(41 76 95)(42 77 96)(43 78 97)(44 79 98)
(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)

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

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

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

132 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 11A···11J12A12B22A···22J22K···22AD33A···33J44A···44T44U···44AN66A···66J132A···132T
order122234444611···11121222···2222···2233···3344···4444···4466···66132···132
size11332113321···1221···13···32···21···13···32···22···2

132 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C11C22C22C22C44S3D6C4×S3S3×C11S3×C22S3×C44
kernelS3×C44C11×Dic3C132S3×C22S3×C11C4×S3Dic3C12D6S3C44C22C11C4C2C1
# reps111141010101040112101020

Matrix representation of S3×C44 in GL2(𝔽397) generated by

1980
0198
,
396396
10
,
10
396396
G:=sub<GL(2,GF(397))| [198,0,0,198],[396,1,396,0],[1,396,0,396] >;

S3×C44 in GAP, Magma, Sage, TeX

S_3\times C_{44}
% in TeX

G:=Group("S3xC44");
// GroupNames label

G:=SmallGroup(264,19);
// by ID

G=gap.SmallGroup(264,19);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-3,226,4404]);
// Polycyclic

G:=Group<a,b,c|a^44=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×C44 in TeX

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