Finite Group Databasebeta

← Prev (#88010) Next (#88012) →

Cayley diagram

D41⋊C5 #88011

order410
isoclinismisoclinic class of D41⋊C5
automorphismAut(G) = AGL(1,𝔽41) (order 1640) · Inn(G) = G · Out(G) = C4
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD41⋊C5
unambiguous semidirect product(C41⋊C5)⋊C2

Splittings

Fitting, socle, derived, SylowC41 ⋊ C2×C5
HallC41⋊C5 ⋊ C2
HallD41 ⋊ C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC41C2×C5
LCSG ▷ C41
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 41

A minimal faithful permutation representation — 2 generators on 41 points:

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

Polycyclic presentation:

2,5,41|c^a=c^40,c^b=c^10

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4223⋊C5, C2×D41⋊C5, C3×D41⋊C5, …
maximal normal subgroup of#4223⋊C5, C2×D41⋊C5, C3×D41⋊C5, …
maximal quotient ofDic41⋊C5, C2×D41⋊C5, C3×D41⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(410, 1);