#72260076

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅²×D₄ ⋊ C₂²
Hall	C₃×C₅² ⋊ H(5,𝔽₂)
Sylow	C₅² ⋊ #1004
Sylow	C₃ ⋊ #11717479

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₅×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅²	C₂⁴
Fitting	C₃×C₅²×D₄	C₂²
hypercenter	D₄	D₅×D₁₅
hypocenter	C₃×C₅²	H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ D₄

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

2^5,3,5^2|a^2=e,c^a=ce,d^a=de,d^b=de,f^d=f^2,g^b=g^4,g^d=g^4,h^b=g^2h,h^d=h^4

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260099
maximal normal subgroup of	#72260099

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