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G = C2×C67⋊C3order 402 = 2·3·67

Direct product of C2 and C67⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C67⋊C3, C134⋊C3, C672C6, SmallGroup(402,2)

Series: Derived Chief Lower central Upper central

C1C67 — C2×C67⋊C3
C1C67C67⋊C3 — C2×C67⋊C3
C67 — C2×C67⋊C3
C1C2

Generators and relations for C2×C67⋊C3
 G = < a,b,c | a2=b67=c3=1, ab=ba, ac=ca, cbc-1=b29 >

67C3
67C6

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 3A3B6A6B67A···67V134A···134V
order12336667···67134···134
size11676767673···33···3

50 irreducible representations

dim111133
type++
imageC1C2C3C6C67⋊C3C2×C67⋊C3
kernelC2×C67⋊C3C67⋊C3C134C67C2C1
# reps11222222

Matrix representation of C2×C67⋊C3 in GL3(𝔽1609) generated by

160800
016080
001608
,
5954781
5964781
5954791
,
57012941585
671544159
40114631104
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

Subgroup lattice of C2×C67⋊C3 in TeX

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