#72222201

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14547427 ⋊ C₂

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₃²	#92301
commutator, cosocle, hypocenter	H(3,𝔽₃)²	C₂×C₃
Frattini	C₃⁴	C₃×C₃⋊D₃
Fitting	#14547427	C₂
LCS	G ▷ H(3,𝔽₃)²
UCS	C₁ ◁ C₃²

802 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 40 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=h,c^a=c^2,c^b=ce,d^a=d^2,d^b=df,e^a=e^2h,e^c=eh,e^d=eg,f^a=f^2g,f^c=fg,f^d=fg

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222317
maximal normal subgroup of	#72222317
maximal quotient of	#72222202, #72222265, #72222270

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