#72277906

order	14880
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×C₃₁ ⋊ C₂×C₃×C₅
Hall	#72277808 ⋊ C₅
Hall	#72277718 ⋊ C₃
Hall	#11758451 ⋊ C₃×C₅
Hall	(C₃₁⋊C₃)⋊C₅ ⋊ #127
Hall	ΓL(1,𝔽₃₂) ⋊ C₃×#127
Hall	C₃₁⋊C₃ ⋊ C₅×#127
Sylow	C₃₁ ⋊ C₃×C₅×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₃₁)
commutator, socle, cosocle	C₂²×C₃₁	C₂³×C₃×C₅
Fitting	C₄²×C₃₁	C₂×C₃×C₅
hypercenter	C₄²	AGL(1,𝔽₃₁)
hypocenter	C₃₁	C₃×C₅×#127
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄²

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 496

Polycyclic presentation:

2^3,3,5,2^2,31|a^2=f,c^2=g,b^a=bf,c^b=cg,h^b=h^30,h^d=h^25,h^e=h^2

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×AGL(1,𝔽₃₁)
maximal normal subgroup of	#127×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", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 2, 2, 31]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^30);;
gap> SetConjugate(coll, 8, 4, F.8^25);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);