#72252811

order	5184
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂³×C₃²

Aliases

No aliases for this group.

Splittings

Sylow

C₂⁶ ⋊ C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	A₄³
commutator, Fitting, socle, cosocle	C₂⁶×C₃	C₃³
prime radical	C₃×A₄×C₂²⋊A₄	C₃
hypocenter	C₂⁶	C₃×3₋¹⁺²
LCS	G ▷ C₂⁶×C₃ ▷ C₂⁶
UCS	C₁ ◁ C₃

895 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 67 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:

3^4,2^6|c^3=d^2,c^a=cd,e^a=ef,f^a=e,g^b=gh,g^c=gh,h^b=g,h^c=g,i^b=ij,i^c=hi,j^b=i,j^c=ghj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×#13049240, #72269067
maximal normal subgroup of	A₄×#13049240, #72269067

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