#72247013

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ #453
Sylow	C₃² ⋊ #89519
Sylow	C₇ ⋊ #10590397

Subgroups

	subgroup	G/H
center	C₂	#72247011
commutator, Fitting, cosocle	C₂×C₃²×C₄×C₇	C₂³
Frattini, hypercenter	C₂×C₄	D₃²×D₇
socle	C₂×C₃²×C₇	#120
hypocenter	C₃²×C₇	#453
LCS	G ▷ C₂×C₃²×C₄×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

780 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^6,3^2,7|b^2=d,e^2=f,b^a=be,c^b=cd,d^a=df,e^a=ef,e^c=ef,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=i^6,i^b=i^6

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247033, D₇×#10590397, #72247112
maximal normal subgroup of	#72247033, D₇×#10590397, #72247112

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