metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D132, C4⋊D33, C3⋊1D44, C33⋊4D4, C44⋊1S3, C11⋊1D12, C132⋊1C2, D66⋊1C2, C12⋊1D11, C2.4D66, C22.10D6, C6.10D22, C66.10C22, sometimes denoted D264 or Dih132 or Dih264, SmallGroup(264,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D132
G = < a,b | a132=b2=1, bab=a-1 >
Smallest permutation representation of D132
►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)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 102)(32 101)(33 100)(34 99)(35 98)(36 97)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)>;
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67) );
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)], [(1,132),(2,131),(3,130),(4,129),(5,128),(6,127),(7,126),(8,125),(9,124),(10,123),(11,122),(12,121),(13,120),(14,119),(15,118),(16,117),(17,116),(18,115),(19,114),(20,113),(21,112),(22,111),(23,110),(24,109),(25,108),(26,107),(27,106),(28,105),(29,104),(30,103),(31,102),(32,101),(33,100),(34,99),(35,98),(36,97),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67)]])
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 66 | 66 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D11 | D12 | D22 | D33 | D44 | D66 | D132 |
| kernel | D132 | C132 | D66 | C44 | C33 | C22 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of D132 ►in GL2(𝔽397) generated by
| 266 | 123 |
| 276 | 153 |
| 182 | 103 |
| 60 | 215 |
G:=sub<GL(2,GF(397))| [266,276,123,153],[182,60,103,215] >;
D132 in GAP, Magma, Sage, TeX
D_{132} % in TeX
G:=Group("D132"); // GroupNames label
G:=SmallGroup(264,25);
// by ID
G=gap.SmallGroup(264,25);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,61,26,323,6004]);
// Polycyclic
G:=Group<a,b|a^132=b^2=1,b*a*b=a^-1>;
// generators/relations
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