#72220655

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×H(3,𝔽₃) ⋊ C₂×D₃
derived	(C₃×A₄)⋊H(3,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	(C₃×A₄)⋊H(3,𝔽₃)	C₂²
Frattini	C₃²	D₃×C₃⋊S₄
Fitting	C₂²×C₃×H(3,𝔽₃)	C₂×D₃
socle	C₂²×C₃²	D₃×C₃⋊D₃
LCS	G ▷ (C₃×A₄)⋊H(3,𝔽₃)
UCS	C₁

1064 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 43 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:

2^2,3^5,2^2|c^a=c^2,c^b=c^2,d^a=d^2,d^c=df,e^a=e^2,e^c=efg^2,f^b=f^2,g^b=g^2,h^a=i,h^d=hi,h^e=i,i^a=h,i^d=h,i^e=hi

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222920, #72222994
maximal normal subgroup of	#72222994
maximal quotient of	#72222920

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