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G = C2×C32  order 64 = 26

Abelian group of type [2,32]

direct product, p-group, abelian, monomial

Aliases: C2×C32, SmallGroup(64,50)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C32
Chief series
C1 — C2 — C4 — C8 — C16 — C2×C16 — C2×C32
Lower central
C1 — C2×C32
Upper central
C1 — C2×C32
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — C2×C32

Generators and relations for C2×C32
 G = < a,b | a2=b32=1, ab=ba >

C1
C2
C2
C2
C4
C4
C22
C8
C2×C4
C8
C2×C8
C16
C16
C32
C32
C2×C16
C2×C32

Smallest permutation representation of C2×C32
►Regular action on 64 points
Generators in S64
(1 63)(2 64)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)

(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)
 
G:=sub<Sym(64)| (1,63)(2,64)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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)>;
 
G:=Group( (1,63)(2,64)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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) );
 
G=PermutationGroup([[(1,63),(2,64),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62)], [(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)]])
 

C2×C32 is a maximal subgroup of
 C32⋊5C4  C22⋊C32  D4.C16  D16⋊2C4  Q32⋊2C4  D16.C4  C4⋊C32  C32⋊3C4  C32⋊4C4  C32.C4  M7(2)  D4○C32  C4○D32  D5⋊C32
C2×C32 is a maximal quotient of
 C22⋊C32  C4⋊C32  M7(2)  D5⋊C32

64 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H16A···16P32A···32AF
order122244448···816···1632···32
size111111111···11···11···1

64 irreducible representations

dim1111111111
type+++
imageC1C2C2C4C4C8C8C16C16C32
kernelC2×C32C32C2×C16C16C2×C8C8C2×C4C4C22C2
# reps12122448832

Matrix representation of C2×C32 ►in GL2(𝔽97) generated by

960
01
,
190
045
G:=sub<GL(2,GF(97))| [96,0,0,1],[19,0,0,45] >;
 

C2×C32 in GAP, Magma, Sage, TeX 

C_2\times C_{32}
 
% in TeX
 
G:=Group("C2xC32");
 
// GroupNames label
 
G:=SmallGroup(64,50);
 
// by ID
 
G=gap.SmallGroup(64,50);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,-2,-2,-2,24,50,69,88]);
 
// Polycyclic
 
G:=Group<a,b|a^2=b^32=1,a*b=b*a>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C32 in TeX

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