#72276291

order	3888
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂⁴×C₃³

Aliases

No aliases for this group.

Splittings

Sylow

C₂⁴ ⋊ #6891

Subgroups

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₂²⋊A₄
commutator, socle, cosocle	C₂⁴×C₃²	C₃³
Fitting, prime radical	C₂⁴×C₃⁴	C₃
hypercenter	C₃⁴	C₂²⋊A₄
hypocenter	C₂⁴	#6891
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃⁴

5122 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 5122 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 108

Polycyclic presentation:

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

Conjugacy classes & character table

456 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, 3, F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);