#72276706

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₃×C₅² ⋊ ΓL(1,𝔽₂₅)
Hall	C₃×C₅²⋊C₃ ⋊ M₁₆
Sylow	C₅² ⋊ #3743

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	C₃×(C₅⋊D₅)⋊C₃	C₂×C₄
Fitting, socle	C₃×C₅²	ΓL(1,𝔽₂₅)
prime radical	((C₅⋊D₅)⋊C₃)⋊D₃	C₄
cosocle	C₃×#1043⋊C₃	C₂²
hypocenter	C₃×C₅²⋊C₃	M₁₆
LCS	G ▷ C₃×(C₅⋊D₅)⋊C₃ ▷ C₃×C₅²⋊C₃
UCS	C₁

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 75

Polycyclic presentation:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AΓL(1,𝔽₂₅)
maximal normal subgroup of	D₃×AΓL(1,𝔽₂₅)

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