#72247669

order	2700
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃³×C₅

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃³×C₅² ⋊ C₄
Sylow	C₃³ ⋊ #1044
Sylow	C₅² ⋊ C₃⋊(C₃⋊Dic₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³×C₅²	C₄
prime radical, cosocle	C₃³×C₅⋊D₅	C₂
LCS	G ▷ C₃³×C₅²
UCS	C₁

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 270

Polycyclic presentation:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247672, C₃⋊(C₃⋊D₃)×#1044, #72247680, …
maximal normal subgroup of	#72247672, C₃⋊(C₃⋊D₃)×#1044, #72247680, …

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