#72246291

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×AGL(1,ℤ/8) ⋊ C₂×C₃
Hall	#72246212 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ #453
Sylow	C₇² ⋊ C₃×#453

Subgroups

	subgroup	G/H
center	C₂	#72246042
commutator, cosocle	C₂×C₄×C₇²	C₂³×C₃
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×AGL(1,ℤ/8)	C₂×C₃
socle	C₂×C₇²	C₃×#120
hypercenter	AGL(1,ℤ/8)	ΓL(1,𝔽₈)⋊D₇
hypocenter	C₇²	C₃×#453
LCS	G ▷ C₂×C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#453×ΓL(1,𝔽₈)⋊D₇, #72246400
maximal normal subgroup of	#453×ΓL(1,𝔽₈)⋊D₇, #72246400

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