#72229915

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×C₅×D₄ ⋊ C₄
Hall	C₃²×C₅ ⋊ #386
Sylow	C₃² ⋊ #65779
Sylow	C₅ ⋊ #10592919

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃²×C₅	C₂²×C₄
Fitting	C₂×C₃²×C₅×D₄	C₄
prime radical	C₂×C₃²×D₄×D₅	C₂
cosocle	C₂²×C₃²×D₅	C₂³
hypercenter	C₂×D₄	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	#386
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

1716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 360

Polycyclic presentation:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×C₃⋊ΓL(1,𝔽₁₆), AGL(1,𝔽₅)×#64146, C₃⋊D₃×#65779
maximal normal subgroup of	#120×C₃⋊ΓL(1,𝔽₁₆), AGL(1,𝔽₅)×#64146, C₃⋊D₃×#65779

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