direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C67⋊C3, C134⋊C3, C67⋊2C6, SmallGroup(402,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C67 — C2×C67⋊C3 |
Generators and relations for C2×C67⋊C3
G = < a,b,c | a2=b67=c3=1, ab=ba, ac=ca, cbc-1=b29 >
Smallest permutation representation of C2×C67⋊C3
►On 134 points
Generators in S134
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134)
(2 38 30)(3 8 59)(4 45 21)(5 15 50)(6 52 12)(7 22 41)(9 29 32)(10 66 61)(11 36 23)(13 43 14)(16 20 34)(17 57 63)(18 27 25)(19 64 54)(24 48 65)(26 55 56)(28 62 47)(31 39 67)(33 46 58)(35 53 49)(37 60 40)(42 44 51)(69 105 97)(70 75 126)(71 112 88)(72 82 117)(73 119 79)(74 89 108)(76 96 99)(77 133 128)(78 103 90)(80 110 81)(83 87 101)(84 124 130)(85 94 92)(86 131 121)(91 115 132)(93 122 123)(95 129 114)(98 106 134)(100 113 125)(102 120 116)(104 127 107)(109 111 118)
G:=sub<Sym(134)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134), (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)(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134), (2,38,30)(3,8,59)(4,45,21)(5,15,50)(6,52,12)(7,22,41)(9,29,32)(10,66,61)(11,36,23)(13,43,14)(16,20,34)(17,57,63)(18,27,25)(19,64,54)(24,48,65)(26,55,56)(28,62,47)(31,39,67)(33,46,58)(35,53,49)(37,60,40)(42,44,51)(69,105,97)(70,75,126)(71,112,88)(72,82,117)(73,119,79)(74,89,108)(76,96,99)(77,133,128)(78,103,90)(80,110,81)(83,87,101)(84,124,130)(85,94,92)(86,131,121)(91,115,132)(93,122,123)(95,129,114)(98,106,134)(100,113,125)(102,120,116)(104,127,107)(109,111,118)>;
G:=Group( (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134), (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)(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134), (2,38,30)(3,8,59)(4,45,21)(5,15,50)(6,52,12)(7,22,41)(9,29,32)(10,66,61)(11,36,23)(13,43,14)(16,20,34)(17,57,63)(18,27,25)(19,64,54)(24,48,65)(26,55,56)(28,62,47)(31,39,67)(33,46,58)(35,53,49)(37,60,40)(42,44,51)(69,105,97)(70,75,126)(71,112,88)(72,82,117)(73,119,79)(74,89,108)(76,96,99)(77,133,128)(78,103,90)(80,110,81)(83,87,101)(84,124,130)(85,94,92)(86,131,121)(91,115,132)(93,122,123)(95,129,114)(98,106,134)(100,113,125)(102,120,116)(104,127,107)(109,111,118) );
G=PermutationGroup([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134)], [(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),(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134)], [(2,38,30),(3,8,59),(4,45,21),(5,15,50),(6,52,12),(7,22,41),(9,29,32),(10,66,61),(11,36,23),(13,43,14),(16,20,34),(17,57,63),(18,27,25),(19,64,54),(24,48,65),(26,55,56),(28,62,47),(31,39,67),(33,46,58),(35,53,49),(37,60,40),(42,44,51),(69,105,97),(70,75,126),(71,112,88),(72,82,117),(73,119,79),(74,89,108),(76,96,99),(77,133,128),(78,103,90),(80,110,81),(83,87,101),(84,124,130),(85,94,92),(86,131,121),(91,115,132),(93,122,123),(95,129,114),(98,106,134),(100,113,125),(102,120,116),(104,127,107),(109,111,118)]])
50 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 67A | ··· | 67V | 134A | ··· | 134V |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 67 | ··· | 67 | 134 | ··· | 134 |
| size | 1 | 1 | 67 | 67 | 67 | 67 | 3 | ··· | 3 | 3 | ··· | 3 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C67⋊C3 | C2×C67⋊C3 |
| kernel | C2×C67⋊C3 | C67⋊C3 | C134 | C67 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 22 | 22 |
Matrix representation of C2×C67⋊C3 ►in GL3(𝔽1609) generated by
| 1608 | 0 | 0 |
| 0 | 1608 | 0 |
| 0 | 0 | 1608 |
| 595 | 478 | 1 |
| 596 | 478 | 1 |
| 595 | 479 | 1 |
| 570 | 1294 | 1585 |
| 67 | 1544 | 159 |
| 401 | 1463 | 1104 |
G:=sub<GL(3,GF(1609))| [1608,0,0,0,1608,0,0,0,1608],[595,596,595,478,478,479,1,1,1],[570,67,401,1294,1544,1463,1585,159,1104] >;
C2×C67⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{67}\rtimes C_3 % in TeX
G:=Group("C2xC67:C3"); // GroupNames label
G:=SmallGroup(402,2);
// by ID
G=gap.SmallGroup(402,2);
# by ID
G:=PCGroup([3,-2,-3,-67,1004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^67=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations
Export