C₃×C₉×D₂₅ #13062446

order	1350
isoclinism	isoclinic class of D₂₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₉×D₂₅
unambiguous semidirect product	C₂₅⋊(C₂×C₃×C₉)

Splittings

Fitting, Hall	C₃×C₉×C₂₅ ⋊ C₂
derived, Sylow	C₂₅ ⋊ C₂×C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₉	D₂₅
commutator, hypocenter	C₂₅	C₂×C₃×C₉
Frattini	C₃×C₅	C₃²×D₅
Fitting	C₃×C₉×C₂₅	C₂
socle	C₃²×C₅	C₃×D₅
prime radical	C₃²×D₂₅	C₃
cosocle	C₃×C₂₅	C₂×C₃²
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₃×C₉

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 675

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

378 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1350, 5);

C3×C9×D25 #13062446