#72276533

order	3942
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃×C₇₃ ⋊ C₂×C₉
Hall	C₃×#10620323 ⋊ C₂
Sylow	C₇₃ ⋊ D₃×C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₇₃	C₂×C₉
prime radical	C₃⋊(D₇₃⋊C₃)	C₃
cosocle	C₃×C₇₃⋊C₃	C₂×C₃
LCS	G ▷ C₃×C₇₃
UCS	C₁

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 219

Polycyclic presentation:

2,3^3,73|b^3=c,d^a=d^2,e^a=e^72,e^b=e^32,e^c=e^64

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#10620323⋊C₂
maximal normal subgroup of	D₃×#10620323⋊C₂

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 73]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> SetConjugate(coll, 5, 2, F.5^32);;
gap> SetConjugate(coll, 5, 3, F.5^64);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);