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G = C2×C68order 136 = 23·17

Abelian group of type [2,68]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C68, SmallGroup(136,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C68
C1C2C34C68 — C2×C68
C1 — C2×C68
C1 — C2×C68

Generators and relations for C2×C68
 G = < a,b | a2=b68=1, ab=ba >


Smallest permutation representation of C2×C68
Regular action on 136 points
Generators in S136
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)

G:=sub<Sym(136)| (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102), (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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)>;

G:=Group( (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102), (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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136) );

G=PermutationGroup([(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,115),(14,116),(15,117),(16,118),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,127),(26,128),(27,129),(28,130),(29,131),(30,132),(31,133),(32,134),(33,135),(34,136),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102)], [(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),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)])

C2×C68 is a maximal subgroup of   C68.4C4  C34.D4  C683C4  D34⋊C4  D685C2

136 conjugacy classes

class 1 2A2B2C4A4B4C4D17A···17P34A···34AV68A···68BL
order1222444417···1734···3468···68
size111111111···11···11···1

136 irreducible representations

dim11111111
type+++
imageC1C2C2C4C17C34C34C68
kernelC2×C68C68C2×C34C34C2×C4C4C22C2
# reps121416321664

Matrix representation of C2×C68 in GL2(𝔽137) generated by

10
0136
,
930
017
G:=sub<GL(2,GF(137))| [1,0,0,136],[93,0,0,17] >;

C2×C68 in GAP, Magma, Sage, TeX

C_2\times C_{68}
% in TeX

G:=Group("C2xC68");
// GroupNames label

G:=SmallGroup(136,9);
// by ID

G=gap.SmallGroup(136,9);
# by ID

G:=PCGroup([4,-2,-2,-17,-2,272]);
// Polycyclic

G:=Group<a,b|a^2=b^68=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C68 in TeX

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