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G = C2xC32order 64 = 26

Abelian group of type [2,32]

direct product, p-group, abelian, monomial

Aliases: C2xC32, SmallGroup(64,50)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2xC32
C1C2C4C8C16C2xC16 — C2xC32
C1 — C2xC32
C1 — C2xC32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2xC32

Generators and relations for C2xC32
 G = < a,b | a2=b32=1, ab=ba >

Subgroups: 17, all normal (13 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, C16, C2xC8, C32, C2xC16, C2xC32

Smallest permutation representation of C2xC32
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)]])

C2xC32 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)  D4oC32  C4oD32  D5:C32
C2xC32 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
kernelC2xC32C32C2xC16C16C2xC8C8C2xC4C4C22C2
# reps12122448832

Matrix representation of C2xC32 in GL2(F97) generated by

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

C2xC32 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 C2xC32 in TeX

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