#72260094

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅² ⋊ M₁₆
Sylow	C₅² ⋊ #3712
Sylow	C₃² ⋊ #87769

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₅²
commutator, socle	C₂×C₃²×C₅²	C₂×C₄
Fitting, cosocle	C₃²×C₄×C₅²	C₂²
prime radical	C₂×C₃⋊(C₃⋊(C₅⋊D₅))	C₄
hypocenter	C₃²×C₅²	M₁₆
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₄

288 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

318 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260100
maximal normal subgroup of	#72260100

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