direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C66, C6⋊C66, C66⋊3C6, C3⋊(C2×C66), (C3×C6)⋊1C22, (C3×C66)⋊4C2, C33⋊4(C2×C6), (C3×C33)⋊9C22, C32⋊2(C2×C22), SmallGroup(396,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C66 |
Generators and relations for S3×C66
G = < a,b,c | a66=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
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)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 131)(19 132)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(65 112)(66 113)
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), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113)>;
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), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113) );
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)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,129),(17,130),(18,131),(19,132),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103),(57,104),(58,105),(59,106),(60,107),(61,108),(62,109),(63,110),(64,111),(65,112),(66,113)]])
198 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33T | 33U | ··· | 33AX | 66A | ··· | 66T | 66U | ··· | 66AX | 66AY | ··· | 66CL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 33 | ··· | 33 | 66 | ··· | 66 | 66 | ··· | 66 | 66 | ··· | 66 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C11 | C22 | C22 | C33 | C66 | C66 | S3 | D6 | C3×S3 | S3×C6 | S3×C11 | S3×C22 | S3×C33 | S3×C66 |
| kernel | S3×C66 | S3×C33 | C3×C66 | S3×C22 | S3×C11 | C66 | S3×C6 | C3×S3 | C3×C6 | D6 | S3 | C6 | C66 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 10 | 20 | 10 | 20 | 40 | 20 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 |
Matrix representation of S3×C66 ►in GL2(𝔽67) generated by
| 61 | 0 |
| 0 | 61 |
| 29 | 0 |
| 30 | 37 |
| 1 | 36 |
| 0 | 66 |
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
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