#72276309

order	3888
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₂⁴ ⋊ #6894

Subgroups

	subgroup	G/H
center, Frattini	C₃²	C₃×A₄²
commutator, socle, cosocle	C₂⁴×C₃²	C₃³
Fitting, prime radical	C₂⁴×C₃³	C₃²
hypercenter	C₃³	A₄²
hypocenter	C₂⁴	#6894
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃³

610 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 162

Polycyclic presentation:

3^5,2^4|b^3=d^2,c^3=e^2,b^a=bd,c^a=cd^2,c^b=ce,f^b=fg,f^c=g,g^b=f,g^c=fg,h^b=gh,h^c=hi,i^b=fi,i^c=h

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1094²
maximal normal subgroup of	#1094²

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