#72245331

order	7056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×D₄ ⋊ C₂×C₃²
Hall	#11716131 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ #41
Sylow	C₇² ⋊ C₃²×#41

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₇²	C₂³×C₃²
Fitting	C₇²×D₄	C₂×C₃²
hypercenter	D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×#41
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ D₄

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#41×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72245389, #72246213
maximal normal subgroup of	#41×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72245389, #72246213

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