#72259772

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₄×#41
Sylow	C₅² ⋊ C₄×#237
Sylow	C₃ ⋊ #13107568

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅²	C₂³×C₄
Fitting	C₂²×C₃×C₅²	C₂²×C₄
prime radical	#72229805	C₂
cosocle	C₂×C₃×C₅⋊D₅	C₂⁴
hypercenter	C₂²	D₃×D₅⋊AGL(1,𝔽₅)
hypocenter	C₃×C₅²	C₄×#41
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13107568
maximal normal subgroup of	D₃×#13107568

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