#72254697

order	5400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	C₅²×3₋¹⁺² ⋊ D₄
Sylow	3₋¹⁺² ⋊ #6038
Sylow	C₅² ⋊ C₄⋊AGL(1,ℤ/9)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅×C₅⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₅²×C₉	C₂²×C₃
Frattini	C₂×C₃	C₃×D₅×D₁₅
Fitting	C₂×C₅²×3₋¹⁺²	C₂²
socle	C₂×C₃×C₅²	C₂×C₃×D₃
hypocenter	C₅²×C₉	C₃×D₄
LCS	G ▷ C₂×C₅²×C₉ ▷ C₅²×C₉
UCS	C₁ ◁ C₂

268 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 180

Polycyclic presentation:

2^2,3,2,3,5^2,3|a^2=d,e^3=h,b^a=bd,e^b=e^2h^2,e^c=eh,f^a=f^4,f^b=f^4,g^a=f^2g,g^b=g^4,h^b=h^2

Conjugacy classes & character table

247 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254701, #72254766, AGL(1,ℤ/9)×#6038
maximal normal subgroup of	#72254701, #72254766, AGL(1,ℤ/9)×#6038

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