#72245379

order	14112
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×M₁₆ ⋊ C₂×C₃²
Hall	#13096548 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ AGL(1,ℤ/8)
Sylow	C₇² ⋊ C₃²×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#72222014
commutator, cosocle	C₄×C₇²	C₂³×C₃²
Frattini	C₄	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
Fitting	C₇²×M₁₆	C₂×C₃²
socle	C₂×C₇²	C₂×C₃²×D₄
hypercenter	M₁₆	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

179 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72246405, #72246407
maximal normal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72246405, #72246407

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