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G = C2×C64order 128 = 27

Abelian group of type [2,64]

direct product, p-group, abelian, monomial

Aliases: C2×C64, SmallGroup(128,159)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C64
C1C2C4C8C16C32C2×C32 — C2×C64
C1 — C2×C64
C1 — C2×C64
C1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4C4C8C8C8C8C16C16C32 — C2×C64

Generators and relations for C2×C64
 G = < a,b | a2=b64=1, ab=ba >


Smallest permutation representation of C2×C64
Regular action on 128 points
Generators in S128
(1 126)(2 127)(3 128)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)
(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)

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

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

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

128 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H16A···16P32A···32AF64A···64BL
order122244448···816···1632···3264···64
size111111111···11···11···11···1

128 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C8C16C16C32C32C64
kernelC2×C64C64C2×C32C32C2×C16C16C2×C8C8C2×C4C4C22C2
# reps121224488161664

Matrix representation of C2×C64 in GL2(𝔽193) generated by

10
0192
,
1190
042
G:=sub<GL(2,GF(193))| [1,0,0,192],[119,0,0,42] >;

C2×C64 in GAP, Magma, Sage, TeX

C_2\times C_{64}
% in TeX

G:=Group("C2xC64");
// GroupNames label

G:=SmallGroup(128,159);
// by ID

G=gap.SmallGroup(128,159);
# by ID

G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,28,58,80,102,124]);
// Polycyclic

G:=Group<a,b|a^2=b^64=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C64 in TeX

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