#72246237

order	2016
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₇×H(5,𝔽₂) ⋊ C₃²
Sylow	H(5,𝔽₂) ⋊ C₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ Ω⁺(4,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	(C₂²⋊A₄)⋊ΓL(1,𝔽₈)
commutator, Fitting, cosocle, hypocenter	C₇×H(5,𝔽₂)	C₃²
socle	C₂×C₇	A₄²
LCS	G ▷ C₇×H(5,𝔽₂)
UCS	C₁ ◁ C₂

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

3^2,2^4,7,2|c^2=h,d^2=h,c^a=cd,c^b=cd,d^a=c,d^b=c,d^c=dh,e^a=deh,e^b=ef,e^d=eh,f^a=cdfh,f^b=e,f^c=fh,g^a=g^2

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×Ω⁺(4,𝔽₃), #72246245
maximal normal subgroup of	ΓL(1,𝔽₈)×Ω⁺(4,𝔽₃), #72246245

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