direct product, abelian, monomial, 3-elementary
Aliases: C3×C42, SmallGroup(126,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C42 |
| C1 — C3×C42 |
| C1 — C3×C42 |
Generators and relations for C3×C42
G = < a,b | a3=b42=1, ab=ba >
Smallest permutation representation of C3×C42
►Regular action on 126 points
Generators in S126
(1 84 87)(2 43 88)(3 44 89)(4 45 90)(5 46 91)(6 47 92)(7 48 93)(8 49 94)(9 50 95)(10 51 96)(11 52 97)(12 53 98)(13 54 99)(14 55 100)(15 56 101)(16 57 102)(17 58 103)(18 59 104)(19 60 105)(20 61 106)(21 62 107)(22 63 108)(23 64 109)(24 65 110)(25 66 111)(26 67 112)(27 68 113)(28 69 114)(29 70 115)(30 71 116)(31 72 117)(32 73 118)(33 74 119)(34 75 120)(35 76 121)(36 77 122)(37 78 123)(38 79 124)(39 80 125)(40 81 126)(41 82 85)(42 83 86)
(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)
G:=sub<Sym(126)| (1,84,87)(2,43,88)(3,44,89)(4,45,90)(5,46,91)(6,47,92)(7,48,93)(8,49,94)(9,50,95)(10,51,96)(11,52,97)(12,53,98)(13,54,99)(14,55,100)(15,56,101)(16,57,102)(17,58,103)(18,59,104)(19,60,105)(20,61,106)(21,62,107)(22,63,108)(23,64,109)(24,65,110)(25,66,111)(26,67,112)(27,68,113)(28,69,114)(29,70,115)(30,71,116)(31,72,117)(32,73,118)(33,74,119)(34,75,120)(35,76,121)(36,77,122)(37,78,123)(38,79,124)(39,80,125)(40,81,126)(41,82,85)(42,83,86), (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)>;
G:=Group( (1,84,87)(2,43,88)(3,44,89)(4,45,90)(5,46,91)(6,47,92)(7,48,93)(8,49,94)(9,50,95)(10,51,96)(11,52,97)(12,53,98)(13,54,99)(14,55,100)(15,56,101)(16,57,102)(17,58,103)(18,59,104)(19,60,105)(20,61,106)(21,62,107)(22,63,108)(23,64,109)(24,65,110)(25,66,111)(26,67,112)(27,68,113)(28,69,114)(29,70,115)(30,71,116)(31,72,117)(32,73,118)(33,74,119)(34,75,120)(35,76,121)(36,77,122)(37,78,123)(38,79,124)(39,80,125)(40,81,126)(41,82,85)(42,83,86), (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) );
G=PermutationGroup([[(1,84,87),(2,43,88),(3,44,89),(4,45,90),(5,46,91),(6,47,92),(7,48,93),(8,49,94),(9,50,95),(10,51,96),(11,52,97),(12,53,98),(13,54,99),(14,55,100),(15,56,101),(16,57,102),(17,58,103),(18,59,104),(19,60,105),(20,61,106),(21,62,107),(22,63,108),(23,64,109),(24,65,110),(25,66,111),(26,67,112),(27,68,113),(28,69,114),(29,70,115),(30,71,116),(31,72,117),(32,73,118),(33,74,119),(34,75,120),(35,76,121),(36,77,122),(37,78,123),(38,79,124),(39,80,125),(40,81,126),(41,82,85),(42,83,86)], [(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)]])
C3×C42 is a maximal subgroup of
C3⋊Dic21
126 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21AV | 42A | ··· | 42AV |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C3×C42 | C3×C21 | C42 | C21 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 8 | 6 | 6 | 48 | 48 |
Matrix representation of C3×C42 ►in GL2(𝔽43) generated by
| 1 | 0 |
| 0 | 6 |
| 20 | 0 |
| 0 | 36 |
G:=sub<GL(2,GF(43))| [1,0,0,6],[20,0,0,36] >;
C3×C42 in GAP, Magma, Sage, TeX
C_3\times C_{42} % in TeX
G:=Group("C3xC42"); // GroupNames label
G:=SmallGroup(126,16);
// by ID
G=gap.SmallGroup(126,16);
# by ID
G:=PCGroup([4,-2,-3,-3,-7]);
// Polycyclic
G:=Group<a,b|a^3=b^42=1,a*b=b*a>;
// generators/relations
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