#72230173

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ #518
Sylow	C₃² ⋊ #66258
Sylow	C₅ ⋊ #10594604

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃²×AGL(1,𝔽₅)
commutator, socle	C₂×C₃²×C₅	C₂³×C₄
Fitting	C₃²×C₄×C₅	C₂²×C₄
prime radical	D₃×D₅×D₁₂	C₂
cosocle	C₂×C₃²×D₅	C₂⁴
hypercenter	C₄	D₃²×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	#518
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

1446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260182, #72260255, AGL(1,𝔽₅)×#64314
maximal normal subgroup of	#72260182, #72260255, AGL(1,𝔽₅)×#64314

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