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Cayley diagram

C2×D41 #4224

order164
isoclinismisoclinic class of D41
automorphismAut(G) = C2×AGL(1,𝔽41) (order 3280) · Inn(G) = D41 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD82
direct productC2×D41
unambiguous semidirect productC41⋊C22
unambiguous semidirect product(C2×C41)⋊C2

Splittings

Fitting, socleC2×C41 ⋊ C2
derived, SylowC41 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D41
commutator, cosocle, hypocenterC41C22
FrattiniC1G
Fitting, socleC2×C41C2
LCSG ▷ C41
UCSC1 ◁ C2

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 82

A minimal faithful permutation representation — 2 generators on 43 points:

(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);(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D41, D164, C22⋊D41, …
maximal normal subgroup ofC4×D41, D164, C22⋊D41, …
maximal quotient ofDic82, C4×D41, D164, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(164, 4);