#72260271

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₄²×C₅ ⋊ C₂×C₄
Hall	C₃²×C₅ ⋊ C₄×#104
Sylow	C₃² ⋊ AGL(1,𝔽₅)×#104
Sylow	C₅ ⋊ #11783877

Subgroups

	subgroup	G/H
center	C₄	#13088706
commutator	C₃²×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂×D₃×ΓL(1,𝔽₁₆)
Fitting	C₃²×C₄²×C₅	C₂×C₄
socle	C₂×C₃²×C₅	C₄×C₂²⋊C₄
prime radical	C₃×D₅×#235	C₂²
cosocle	C₂×C₃²×C₄×D₅	C₂³
hypercenter	C₄²	D₃×ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	C₄×#104
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

2^7,3^2,5|a^2=g,b^2=e,c^2=d,f^2=g,b^a=bf,e^a=eg,f^a=fg,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2,j^c=j^3,j^d=j^4

Conjugacy classes & character table

234 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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);