#72243888

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₃×#6908 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×#6908	C₂²
Frattini	C₃²	D₃×C₃⋊(C₃⋊D₃)
socle	C₃²	((C₃⋊D₃)⋊C₃)⋊D₃
LCS	G ▷ C₃×#6908
UCS	C₁

1024 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 81

Polycyclic presentation:

2^2,3^6|e^3=h^2,c^b=c^2,d^a=d^2,d^c=dg,e^b=e^2h,e^d=eg^2,f^a=fg^2h^2,f^b=f^2,f^d=fh^2,g^a=g^2h,g^b=g^2,g^d=gh,h^b=h^2

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243897, #72243915, #72243919, …
maximal normal subgroup of	#72243897, #72243915, #72243919, …

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