#72247706

order	2700
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃³×C₅² ⋊ C₂²
Sylow	C₃³ ⋊ D₅²
Sylow	C₅² ⋊ D₃×C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₅²	C₂²
LCS	G ▷ C₃³×C₅²
UCS	C₁

368 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 61 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:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11744719, D₅×C₃⋊D₃×D₁₅, D₅×D₁₅⋊(C₃⋊D₃)
maximal normal subgroup of	D₃×#11744719, D₅×C₃⋊D₃×D₁₅, D₅×D₁₅⋊(C₃⋊D₃)

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