#72222307

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14555226 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#6908	C₂×C₃²
Frattini	C₃×C₉	C₃²×C₃⋊D₃
Fitting	#14555226	C₂
socle	C₃	C₃⋊#92285
LCS	G ▷ #6908
UCS	C₁

576 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^7|c^3=g^2h^2,d^3=h,f^3=h,c^b=cg^2h^2,d^a=d^2h^2,d^c=df,e^a=e^2,e^d=eg,f^a=f^2g^2h,f^b=fh,f^d=fg^2h,g^a=gh^2,g^d=gh,h^a=h^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72222308

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