#72244951

order	2592
automorphism	?
homology	Abelianization (H₁) = C₉² · Schur multiplier (H₂) = C₂×C₉

solvable

Aliases

No aliases for this group.

Splittings

derived, Sylow

H(5,𝔽₂) ⋊ C₉²

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂×C₃²	A₄²
commutator, hypocenter	H(5,𝔽₂)	C₉²
Fitting, prime radical, cosocle	C₃²×H(5,𝔽₂)	C₃²
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃²

231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 648

Polycyclic presentation:

3^4,2^5|a^3=c,b^3=d,e^2=i,f^2=i,e^a=ef,e^b=f,f^a=e,f^b=ef,f^e=fi,g^a=fgi,g^b=gh,g^f=gi,h^a=ehi,h^b=g,h^e=hi,h^f=hi

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72245426, #72252975, C₂×#72245426
maximal subgroup of	#72245041, C₂×#72244951, #72245426, …
maximal normal subgroup of	#72245041, C₂×#72244951, #72245426, …
maximal quotient of	C₂×#72244951

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