#72230868

order	3720
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₂²×C₃₁ ⋊ C₂×C₃×C₅
Hall	C₂²×D₃₁⋊C₅ ⋊ C₃
Hall	D₃₁⋊A₄ ⋊ C₅
Hall	C₂²×D₃₁ ⋊ C₃×C₅
Hall	ΓL(1,𝔽₃₂) ⋊ C₂×A₄
Sylow	C₃₁ ⋊ C₂×C₅×A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₃₁	C₂×C₃×C₅
LCS	G ▷ C₂²×C₃₁
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 124

Polycyclic presentation:

2,3,5,2^2,31|d^b=de,e^b=d,f^a=f^30,f^b=f^25,f^c=f^16

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×AGL(1,𝔽₃₁)
maximal normal subgroup of	A₄×AGL(1,𝔽₃₁)

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, 3, 5, 2, 2, 31]);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.4);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 6, 2, F.6^25);;
gap> SetConjugate(coll, 6, 3, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);