#72246294

order	14112
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Hall	#13096150 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ #99
Sylow	C₇² ⋊ C₃²×#99

Subgroups

	subgroup	G/H
center	C₂	#72246092
commutator	C₂²×C₇²	C₂×C₃²×C₄
Frattini	C₂³	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
Fitting	C₂×C₇²×D₄	C₂×C₃²
socle	C₂×C₇²	C₃²×C₂²⋊C₄
prime radical	C₂×D₄×ΓL(1,𝔽₈)²	C₂
cosocle	C₂³×C₇²	C₂²×C₃²
hypercenter	C₂×D₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×#99
LCS	G ▷ C₂²×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

179 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#99×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72246308, #72246319
maximal normal subgroup of	#99×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72246308, #72246319

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