#72259497

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₅² ⋊ D₃×AGL(1,ℤ/8)
Sylow	C₃ ⋊ #13106298

Subgroups

	subgroup	G/H
center	C₂	D₃×#87886
commutator, cosocle	C₃×C₄×C₅²	C₂⁴
Frattini	C₄	C₂×D₃×D₅²
Fitting	C₃×C₅²×D₄	C₂³
socle	C₂×C₃×C₅²	C₂²×D₄
hypercenter	D₄	D₃×D₅²
hypocenter	C₃×C₅²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13106298
maximal normal subgroup of	D₃×#13106298

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