#72258594

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#6342371 ⋊ C₃×D₃
derived	C₂⁴×C₂²⋊A₄ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴×C₂²⋊A₄	C₂²×C₃
Frattini, socle	C₂⁴	C₂×A₄×S₄
Fitting	#6342371	C₃×D₃
LCS	G ▷ C₂⁴×C₂²⋊A₄
UCS	C₁

At least 7619 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has at least 7619 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:

2^2,3,2^4,3,2^4|d^a=df,d^c=e,e^a=eg,e^c=de,f^c=g,g^c=fg,h^b=h^2,i^a=ik,i^b=ij,i^h=j,j^a=jkl,j^h=ij,k^b=l,k^h=kl,l^b=k,l^h=k

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258869
maximal normal subgroup of	#72258869

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> SetConjugate(coll, 11, 2, F.12);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
gap> SetConjugate(coll, 12, 2, F.11);;
gap> SetConjugate(coll, 12, 8, F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);