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G = C2×C32order 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

C1 — C2×C32
C1C2C4C8C16C2×C16 — C2×C32
C1 — C2×C32
C1 — C2×C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2×C32

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


Smallest permutation representation of C2×C32
Regular action on 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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([(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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
C325C4  C22⋊C32  D4.C16  D162C4  Q322C4  D16.C4  C4⋊C32  C323C4  C324C4  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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