#72230455

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ C₂×#417
Sylow	C₃² ⋊ C₂×#65793
Sylow	C₅ ⋊ #11857908

Subgroups

	subgroup	G/H
center	C₂	C₂×#13088405
commutator	C₃²×C₄×C₅	C₂³×C₄
Frattini	C₄	C₂×D₃²×AGL(1,𝔽₅)
Fitting	C₂×C₃²×C₄×C₅	C₂²×C₄
socle	C₂×C₃²×C₅	C₂²×C₂²⋊C₄
prime radical	D₅×#64306	C₂
cosocle	C₃²×C₄×D₅	C₂⁴
hypercenter	C₂×C₄	D₃²×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	C₂×#417
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄

3682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#10595100
maximal normal subgroup of	AGL(1,𝔽₅)×#10595100

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