#72246407

order	28224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×#100 ⋊ C₂×C₃²
Hall	#72246218 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ #453
Sylow	C₇² ⋊ C₃²×#453

Subgroups

	subgroup	G/H
center	C₂	#72246104
commutator, cosocle	C₂×C₄×C₇²	C₂³×C₃²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
Fitting	C₇²×#100	C₂×C₃²
socle	C₂×C₇²	C₃²×#120
hypercenter	#100	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×#453
LCS	G ▷ C₂×C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

2352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^3,3^2,2^3,7^2|g^2=h,b^a=bg,c^a=ch,c^b=cf,f^a=fh,g^a=gh,g^b=gh,g^c=gh,i^a=i^6,i^b=i^6,i^d=j,i^e=i^2,j^a=j^6,j^b=j^6,j^d=i^5j^3,j^e=j^2

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#453×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
maximal normal subgroup of	#453×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 7, 7]);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^6);;
gap> SetConjugate(coll, 10, 2, F.10^6);;
gap> SetConjugate(coll, 10, 4, F.9^5*F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);