#72222936

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃³ ⋊ C₂×C₃×D₃
derived	C₂²×C₃×C₂²⋊H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×C₃×C₂²⋊H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	D₃×A₄×S₄
Fitting	C₂⁴×C₃³	C₂×C₃×D₃
socle	C₂⁴×C₃²	C₃×D₃²
LCS	G ▷ C₂²×C₃×C₂²⋊H(3,𝔽₃)
UCS	C₁

4215 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 4215 conjugacy classes of subgroups — too many to draw. Only its 57 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:

2^2,3,2^2,3^4,2^2|d^c=e,e^c=de,f^b=f^2,f^c=fh,g^a=g^2,g^f=gi,h^b=h^2,i^a=i^2,i^b=i^2,j^g=k,k^a=jk,k^g=jk

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226353
maximal normal subgroup of	#72226353

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 2]);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 11, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);