#72222263

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14551589 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(#6882⋊C₂)
commutator, cosocle, hypocenter	#10624733	C₂×C₃
Frattini	C₃²×C₉	C₃×C₃⋊D₃
Fitting	#14551589	C₂
socle	C₃²	C₃⋊#4178
prime radical	#13091056	C₃
LCS	G ▷ #10624733
UCS	C₁ ◁ C₃

439 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 35 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|b^3=gh^2,c^3=h^2,d^3=h^2,e^3=h^2,c^a=c^2h,c^b=ce,d^a=d^2h,d^b=dh^2,d^c=df,e^a=e^2g,e^b=eh^2,e^c=eg,f^a=fh^2,f^c=fh,g^a=gh,g^c=gh,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222265, #72222315
maximal normal subgroup of	#72222315

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