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Cayley diagram

D164 #66496

order328
isoclinismisoclinic class of Dic82
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD164
unambiguous semidirect productC4⋊D41

Splittings

FittingC4×C41 ⋊ C2
SylowC41 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D41
commutator, socle, cosocleC2×C41C22
FittingC4×C41C2
hypercenterC4D41
hypocenterC41D4
LCSG ▷ C2×C41 ▷ C41
UCSC1 ◁ C2 ◁ C4

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 164

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

2^3,41|b^2=c,b^a=bc,d^a=d^40

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620273, D328, #10620282, …
maximal normal subgroup of#10620273, D328, #10620282, …
maximal quotient of#10620273, D328, Dic164, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(328, 6);