#72243953

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×C₉⋊H(3,𝔽₃) ⋊ C₂²
derived	C₃³×C₉ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃
Frattini	C₃²	C₃×D₃×C₃⋊D₃
Fitting	C₃×C₉⋊H(3,𝔽₃)	C₂²
socle	C₃³	C₃×D₃²
LCS	G ▷ C₃³×C₉
UCS	C₁

728 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 162

Polycyclic presentation:

2^2,3^6|e^3=h,d^a=d^2,d^c=dg,e^b=e^2h^2,e^c=eh,f^b=f^2,g^a=g^2,h^b=h^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243962, #72243965, (C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9), …
maximal normal subgroup of	(C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9), D₃×D₉⋊((C₃⋊D₃)⋊C₃)
maximal quotient of	#72243962, #72243965

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