#72247752

order	5400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃³×C₅² ⋊ C₂×C₄
Sylow	C₃³ ⋊ D₅⋊AGL(1,𝔽₅)
Sylow	C₅² ⋊ #6289

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³×C₅²	C₂×C₄
prime radical	#72247642	C₂
cosocle	C₃³×C₅⋊D₅	C₂²
LCS	G ▷ C₃³×C₅²
UCS	C₁

648 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#10597182, #72247781, D₃×#14268426
maximal normal subgroup of	C₃⋊D₃×#10597182, #72247781, D₃×#14268426

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