#72259630

order	9600
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₄×AGL(1,ℤ/8)
Sylow	C₅² ⋊ #79773
Sylow	C₃ ⋊ #72213282

Subgroups

	subgroup	G/H
center	C₂	D₃×#11717375
commutator	C₃×C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂×D₃×D₅⋊AGL(1,𝔽₅)
Fitting	C₃×C₅²×D₄	C₂²×C₄
socle	C₂×C₃×C₅²	C₂×C₄×D₄
prime radical	#72230898	C₂
cosocle	C₃×C₄×C₅⋊D₅	C₂⁴
hypercenter	D₄	D₃×D₅⋊AGL(1,𝔽₅)
hypocenter	C₃×C₅²	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

3044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213282
maximal normal subgroup of	D₃×#72213282

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