#72252823

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	C₃	C₃×#10596782
commutator, socle, cosocle	C₂⁶×C₃	C₃³
Fitting	C₂⁶×C₃²	C₃²
prime radical	C₂²×C₃²×C₂²⋊A₄	C₃
hypercenter	C₃²	#10596782
hypocenter	C₂⁶	C₃×3₋¹⁺²
LCS	G ▷ C₂⁶×C₃ ▷ C₂⁶
UCS	C₁ ◁ C₃ ◁ C₃²

2596 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 2596 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 54

Polycyclic presentation:

3^4,2^6|c^3=d^2,c^a=cd,e^c=ef,f^c=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

136 conjugacy classes — view the full character table.

Relationships

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

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, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, 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);