#72243909

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#10624760 ⋊ C₂²
derived	C₃×#731 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×#731	C₂²×C₃
Frattini	C₃³	C₃×D₃²
Fitting	#10624760	C₂²
socle	C₃	(((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃
LCS	G ▷ C₃×#731
UCS	C₁

289 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 27 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|d^3=h^2,d^b=d^2h,d^c=df,e^a=e^2,e^d=eg,f^b=f^2,f^c=fh^2,g^a=g^2h,g^b=g^2,g^e=gh,h^b=h^2

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243915, #72243939, #72244028
maximal normal subgroup of	#72243939, #72244028

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