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Cayley diagram

Dic41 #4221

order164
isoclinismisoclinic class of D41
automorphismAut(G) = C2×AGL(1,𝔽41) (order 3280) · Inn(G) = D41 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic41

Splittings

derived, SylowC41 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D41
commutator, hypocenterC41C4
Fitting, socle, prime radical, cosocleC2×C41C2
LCSG ▷ C41
UCSC1 ◁ C2

6 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);(1,3,2,4)(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:

2^2,41|a^2=b,c^a=c^40

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66493, Dic82, C4×D41, …
maximal normal subgroup of#66493, Dic82, C4×D41, …
maximal quotient of#66491, C2×Dic41, C3×Dic41, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(164, 1);