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Cayley diagram

C3×D25 #3871

order150
isoclinismisoclinic class of D25
automorphismAut(G) = C2×AGL(1,ℤ/25) (order 1000) · Inn(G) = D25 · Out(G) = C22×C5
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D25
unambiguous semidirect productC25⋊(C2×C3)

Splittings

Fitting, HallC3×C25 ⋊ C2
derived, SylowC25 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3D25
commutator, cosocle, hypocenterC25C2×C3
FrattiniC5C3×D5
FittingC3×C25C2
socleC3×C5D5
LCSG ▷ C25
UCSC1 ◁ C3

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)(5,7)(6,10)(8,13)(9,16)(11,19)(12,18)(14,21)(15,23)(17,25)(20,26)(22,27)(24,28);(4,9,24,23,7,6,17,27,13,12,11,26,21,14,20,19,18,8,22,25,10,5,15,28,16)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#1035, C3⋊#1035, D3×D25, …
maximal normal subgroup ofC3×#1035, C3⋊#1035, D3×D25, …
maximal quotient ofC3×Dic25, C2×C3×D25, C9×D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 2);