direct product, abelian, monomial, 2-elementary
Aliases: C2×C66, SmallGroup(132,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C66 |
| C1 — C2×C66 |
| C1 — C2×C66 |
Generators and relations for C2×C66
G = < a,b | a2=b66=1, ab=ba >
Smallest permutation representation of C2×C66
►Regular action on 132 points
Generators in S132
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)
(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)
G:=sub<Sym(132)| (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117), (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)>;
G:=Group( (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117), (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) );
G=PermutationGroup([[(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117)], [(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)]])
C2×C66 is a maximal subgroup of
C33⋊7D4
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 11A | ··· | 11J | 22A | ··· | 22AD | 33A | ··· | 33T | 66A | ··· | 66BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 |
| kernel | C2×C66 | C66 | C2×C22 | C22 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 10 | 30 | 20 | 60 |
Matrix representation of C2×C66 ►in GL2(𝔽67) generated by
| 66 | 0 |
| 0 | 1 |
| 36 | 0 |
| 0 | 41 |
G:=sub<GL(2,GF(67))| [66,0,0,1],[36,0,0,41] >;
C2×C66 in GAP, Magma, Sage, TeX
C_2\times C_{66} % in TeX
G:=Group("C2xC66"); // GroupNames label
G:=SmallGroup(132,10);
// by ID
G=gap.SmallGroup(132,10);
# by ID
G:=PCGroup([4,-2,-2,-3,-11]);
// Polycyclic
G:=Group<a,b|a^2=b^66=1,a*b=b*a>;
// generators/relations
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