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Cayley diagram

D75 #3872

order150
isoclinismisoclinic class of D75
automorphismAut(G) = D3×AGL(1,ℤ/25) (order 3000) · Inn(G) = G · Out(G) = C4×C5
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD75
unambiguous semidirect productC3⋊D25
unambiguous semidirect productC25⋊D3

Splittings

Fitting, derived, HallC3×C25 ⋊ C2
SylowC25 ⋊ D3
SylowC3 ⋊ D25

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C25C2
FrattiniC5D15
socleC3×C5D5
LCSG ▷ C3×C25
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 75

A minimal faithful permutation representation — 2 generators on 28 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);(2,3)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)

Polycyclic presentation:

2,3,5^2|c^5=d^4,b^a=b^2,c^a=c^4d,d^a=d^4

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D25, C2×D75, D225, …
maximal normal subgroup ofD3×D25, C2×D75, C3×D75, …
maximal quotient ofDic75, C2×D75, D225, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 3);