#72218430

order	6720
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	#72218427 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ #385
Hall	C₅×C₇ ⋊ C₃×#385
Hall	ΓL(1,𝔽₈) ⋊ #65699
Sylow	C₇ ⋊ C₃×#65699
Sylow	C₅ ⋊ #89718⋊C₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇)
commutator, socle	C₂²×C₅×C₇	C₂²×C₃×C₄
Fitting	C₂³×C₅×C₇	C₂×C₃×C₄
prime radical	C₂³×D₅×ΓL(1,𝔽₈)	C₂²
cosocle	C₂²×C₇×D₅	C₂³×C₃
hypercenter	C₂³	AGL(1,𝔽₅)×AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₃×#385
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,5,7|b^2=e,c^2=ef,c^a=cf,c^b=cfg,h^b=h^3,h^c=h^3,h^e=h^4,i^b=i^6,i^d=i^2

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218606, AGL(1,𝔽₇)×#65699, AGL(1,𝔽₅)×#6500⋊C₃
maximal normal subgroup of	#72218606, AGL(1,𝔽₇)×#65699, AGL(1,𝔽₅)×#6500⋊C₃

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 5, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 3, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^4);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);