#72245804

order	4704
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×D₄ ⋊ C₂²×C₃
Hall	#13096696 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ H(5,𝔽₂)
Sylow	C₇² ⋊ C₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₇×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₇²	C₂⁴×C₃
Fitting	C₇²×D₄	C₂²×C₃
hypercenter	D₄	D₇×AGL(1,𝔽₇)
hypocenter	C₇²	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ D₄

691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^4,3,2,7^2|a^2=f,d^2=f,b^a=bf,c^a=cf,d^a=df,d^b=df,g^a=g^6,g^c=g^6,g^e=g^4,h^a=g^6h,h^c=h^6,h^e=g^5h

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245862
maximal normal subgroup of	#72245862

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