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Cayley diagram

C2×C3×D25 #64466

order300
isoclinismisoclinic class of D25
automorphismAut(G) = C22×AGL(1,ℤ/25) (order 2000) · Inn(G) = D25 · Out(G) = C23×C5
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×D25
unambiguous semidirect productC25⋊(C22×C3)
unambiguous semidirect product(C2×C25)⋊(C2×C3)

Splittings

FittingC2×C3×C25 ⋊ C2
derived, SylowC25 ⋊ C22×C3
HallC3×C25 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2×C3D25
commutator, cosocle, hypocenterC25C22×C3
FrattiniC5C2×C3×D5
FittingC2×C3×C25C2
socleC2×C3×C5D5
LCSG ▷ C25
UCSC1 ◁ C2×C3

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 150

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2)(3,4,5)(7,9)(8,12)(10,15)(11,18)(13,21)(14,20)(16,23)(17,25)(19,27)(22,28)(24,29)(26,30);(1,2)(6,11,26,25,9,8,19,29,15,14,13,28,23,16,22,21,20,10,24,27,12,7,17,30,18)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×D25, #10597029, #10597031, …
maximal normal subgroup ofDic3×D25, #10597029, #10597031, …
maximal quotient ofC3×Dic50, C3×C4×D25, C3×D100, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 9);