#72230156

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ C₂×#99
Sylow	C₃² ⋊ D₅⋊#99
Sylow	C₅ ⋊ C₂×#63581

Subgroups

	subgroup	G/H
center	C₂	C₂×#10623978
commutator	C₂²×C₃²×C₅	C₂²×C₄
Frattini	C₂²	C₂×ΓL(1,𝔽₁₆)⋊D₃
Fitting	C₂³×C₃²×C₅	C₂×C₄
socle	C₂×C₃²×C₅	C₂×C₂²⋊C₄
prime radical	C₂×D₅×#638	C₂
cosocle	C₂²×C₃²×D₅	C₂³
hypercenter	C₂³	ΓL(1,𝔽₁₆)⋊D₃
hypocenter	C₃²×C₅	C₂×#99
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂³

1076 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 131 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:

2^6,3^2,5|a^2=d,b^a=be,c^b=cf,d^b=df,e^a=ef,g^a=g^2,g^b=g^2,h^a=g^2h,h^b=h^2,i^a=i^3,i^d=i^4

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);