#72218342

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₇×D₄ ⋊ C₃×C₄
Hall	#72218341 ⋊ C₃
Hall	C₅×ΓL(1,𝔽₈) ⋊ #418
Hall	C₅×C₇ ⋊ C₃×#418
Hall	ΓL(1,𝔽₈) ⋊ #65767
Sylow	C₇ ⋊ C₃×#65767
Sylow	C₅ ⋊ #89987⋊C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×(C₂²⋊(C₇⋊AGL(1,𝔽₅)))⋊C₃
commutator	C₄×C₅×C₇	C₂²×C₃×C₄
Frattini	C₄	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
Fitting	C₂×C₅×C₇×D₄	C₃×C₄
socle	C₂×C₅×C₇	C₂×C₃×C₂²⋊C₄
prime radical	C₂×D₄×D₅×ΓL(1,𝔽₈)	C₂
cosocle	C₄×C₇×D₅	C₂³×C₃
hypercenter	C₂×D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₃×#418
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,5,7|a^2=e,e^2=g,f^2=g,b^a=bf,c^a=cg,e^b=eg,f^b=fg,h^a=h^3,h^e=h^4,i^a=i^6,i^d=i^2

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×#6495⋊C₃, AGL(1,𝔽₇)×#65767
maximal normal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×#6495⋊C₃, AGL(1,𝔽₇)×#65767

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