#72244993

order	2592
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂×C₃

solvable

Aliases

No aliases for this group.

Splittings

Sylow

H(5,𝔽₂) ⋊ C₉⋊C₉

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂×C₃²	A₄²
commutator	C₃×H(5,𝔽₂)	C₃×C₉
Fitting, prime radical, cosocle	C₃²×H(5,𝔽₂)	C₃²
hypocenter	H(5,𝔽₂)	C₉⋊C₉
LCS	G ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃²

201 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 648

Polycyclic presentation:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245041, C₂×#72244993, #72252993, …
maximal normal subgroup of	#72245041, C₂×#72244993, #72252993, …
maximal quotient of	C₂×#72244993, #72276108

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.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 5, 1, F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.5*F.8*F.9);;
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);