#72220357

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃²×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃⁴	C₂⁴
Fitting	C₂²×C₃⁴	C₂³
hypercenter	C₂²	D₃²×C₃⋊D₃
hypocenter	C₃⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

3466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

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