direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C44, D6.C22, C132⋊6C2, C12⋊2C22, C22.14D6, Dic3⋊2C22, C66.19C22, C3⋊1(C2×C44), C33⋊6(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
| C3 — S3×C44 |
Generators and relations for S3×C44
G = < a,b,c | a44=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 46 109)(2 47 110)(3 48 111)(4 49 112)(5 50 113)(6 51 114)(7 52 115)(8 53 116)(9 54 117)(10 55 118)(11 56 119)(12 57 120)(13 58 121)(14 59 122)(15 60 123)(16 61 124)(17 62 125)(18 63 126)(19 64 127)(20 65 128)(21 66 129)(22 67 130)(23 68 131)(24 69 132)(25 70 89)(26 71 90)(27 72 91)(28 73 92)(29 74 93)(30 75 94)(31 76 95)(32 77 96)(33 78 97)(34 79 98)(35 80 99)(36 81 100)(37 82 101)(38 83 102)(39 84 103)(40 85 104)(41 86 105)(42 87 106)(43 88 107)(44 45 108)
(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), (1,46,109)(2,47,110)(3,48,111)(4,49,112)(5,50,113)(6,51,114)(7,52,115)(8,53,116)(9,54,117)(10,55,118)(11,56,119)(12,57,120)(13,58,121)(14,59,122)(15,60,123)(16,61,124)(17,62,125)(18,63,126)(19,64,127)(20,65,128)(21,66,129)(22,67,130)(23,68,131)(24,69,132)(25,70,89)(26,71,90)(27,72,91)(28,73,92)(29,74,93)(30,75,94)(31,76,95)(32,77,96)(33,78,97)(34,79,98)(35,80,99)(36,81,100)(37,82,101)(38,83,102)(39,84,103)(40,85,104)(41,86,105)(42,87,106)(43,88,107)(44,45,108), (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), (1,46,109)(2,47,110)(3,48,111)(4,49,112)(5,50,113)(6,51,114)(7,52,115)(8,53,116)(9,54,117)(10,55,118)(11,56,119)(12,57,120)(13,58,121)(14,59,122)(15,60,123)(16,61,124)(17,62,125)(18,63,126)(19,64,127)(20,65,128)(21,66,129)(22,67,130)(23,68,131)(24,69,132)(25,70,89)(26,71,90)(27,72,91)(28,73,92)(29,74,93)(30,75,94)(31,76,95)(32,77,96)(33,78,97)(34,79,98)(35,80,99)(36,81,100)(37,82,101)(38,83,102)(39,84,103)(40,85,104)(41,86,105)(42,87,106)(43,88,107)(44,45,108), (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)], [(1,46,109),(2,47,110),(3,48,111),(4,49,112),(5,50,113),(6,51,114),(7,52,115),(8,53,116),(9,54,117),(10,55,118),(11,56,119),(12,57,120),(13,58,121),(14,59,122),(15,60,123),(16,61,124),(17,62,125),(18,63,126),(19,64,127),(20,65,128),(21,66,129),(22,67,130),(23,68,131),(24,69,132),(25,70,89),(26,71,90),(27,72,91),(28,73,92),(29,74,93),(30,75,94),(31,76,95),(32,77,96),(33,78,97),(34,79,98),(35,80,99),(36,81,100),(37,82,101),(38,83,102),(39,84,103),(40,85,104),(41,86,105),(42,87,106),(43,88,107),(44,45,108)], [(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 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 11A | ··· | 11J | 12A | 12B | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33J | 44A | ··· | 44T | 44U | ··· | 44AN | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C44 | S3 | D6 | C4×S3 | S3×C11 | S3×C22 | S3×C44 |
| kernel | S3×C44 | C11×Dic3 | C132 | S3×C22 | S3×C11 | C4×S3 | Dic3 | C12 | D6 | S3 | C44 | C22 | C11 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 40 | 1 | 1 | 2 | 10 | 10 | 20 |
Matrix representation of S3×C44 ►in GL2(𝔽397) generated by
| 198 | 0 |
| 0 | 198 |
| 396 | 396 |
| 1 | 0 |
| 1 | 0 |
| 396 | 396 |
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