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G = C82order 64 = 26

Abelian group of type [8,8]

direct product, p-group, abelian, monomial

Aliases: C82, SmallGroup(64,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C82
C1C2C22C2×C4C42C4×C8 — C82
C1 — C82
C1 — C82
C1C22C22C42 — C82

Generators and relations for C82
 G = < a,b | a8=b8=1, ab=ba >


Smallest permutation representation of C82
Regular action on 64 points
Generators in S64
(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)
(1 42 34 51 23 12 57 25)(2 43 35 52 24 13 58 26)(3 44 36 53 17 14 59 27)(4 45 37 54 18 15 60 28)(5 46 38 55 19 16 61 29)(6 47 39 56 20 9 62 30)(7 48 40 49 21 10 63 31)(8 41 33 50 22 11 64 32)

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

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

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

C82 is a maximal subgroup of
C165C8  C8⋊C16  C8≀C2  C82C16  C8.36D8  C82⋊C2  C8215C2  C822C2  C86M4(2)  C86D8  C89SD16  C86Q16  C88SD16  C85D8  C85Q16  C8212C2  C825C2  C8.7Q16  C823C2  C84D8  C84Q16  C82⋊C3
C82 is a maximal quotient of
C2.C82  C165C8

64 conjugacy classes

class 1 2A2B2C4A···4L8A···8AV
order12224···48···8
size11111···11···1

64 irreducible representations

dim1111
type++
imageC1C2C4C8
kernelC82C4×C8C2×C8C8
# reps131248

Matrix representation of C82 in GL2(𝔽17) generated by

80
013
,
130
015
G:=sub<GL(2,GF(17))| [8,0,0,13],[13,0,0,15] >;

C82 in GAP, Magma, Sage, TeX

C_8^2
% in TeX

G:=Group("C8^2");
// GroupNames label

G:=SmallGroup(64,2);
// by ID

G=gap.SmallGroup(64,2);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,24,55,86,117]);
// Polycyclic

G:=Group<a,b|a^8=b^8=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C82 in TeX

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