#72220526

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂²×C₃³

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×C₃³ ⋊ C₃×D₃

Subgroups

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

1310 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 50 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:

2,3^3,2^2,3^2,2^2|c^b=cd,e^c=f,f^c=ef,g^a=g^2,g^b=gh,h^a=h^2,i^g=j,j^a=ij,j^g=ij

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222936, #72222957, #72222968
maximal normal subgroup of	#72222936, #72222957, #72222968

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