#72247888

order	3348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃²×C₃₁ ⋊ C₂²×C₃
Hall	C₃²×C₃₁⋊C₃ ⋊ C₂²
Sylow	C₃₁ ⋊ C₃×D₃²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₃₁	C₂²×C₃
LCS	G ▷ C₃²×C₃₁
UCS	C₁

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 186

Polycyclic presentation:

2^2,3^3,31|d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^30,f^c=f^5

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×D₃₁⋊C₃, #72247908
maximal normal subgroup of	D₃²×D₃₁⋊C₃, #72247908

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 31]);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 6, 3, F.6^5);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);