#72246901

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ C₂×#120
Sylow	C₃² ⋊ C₂×#6516
Sylow	C₇ ⋊ #10595278

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃²×D₇
commutator, socle, cosocle	C₂²×C₃²×C₇	C₂⁴
Fitting	C₂³×C₃²×C₇	C₂³
hypercenter	C₂³	D₃²×D₇
hypocenter	C₃²×C₇	C₂×#120
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

2884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 336

Polycyclic presentation:

2^6,3^2,7|b^2=e,b^a=bf,c^b=ce,g^a=g^2,g^d=g^2,h^a=g^2h,h^d=h^2,i^b=i^6

Conjugacy classes & character table

282 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#10595278
maximal normal subgroup of	D₇×#10595278
maximal quotient of	#72246909

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