#72243784

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#10624935 ⋊ C₂²
derived	#6908 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#6908	C₂²×C₃
Frattini	C₃²	C₃×D₃×C₃⋊D₃
Fitting	#10624935	C₂²
socle	C₃	C₃×((C₃⋊D₃)⋊C₃)⋊D₃
LCS	G ▷ #6908
UCS	C₁

483 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 49 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|f^3=h^2,c^a=cg,d^a=de,d^b=d^2,e^a=e^2,e^b=e^2gh,e^c=eh^2,e^d=eg,f^b=f^2h,f^d=fg^2h,f^e=fg^2h,g^a=g^2,g^d=gh,g^e=gh,h^b=h^2

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243786, #72243856
maximal normal subgroup of	#72243856

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