#72243908

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#10624930 ⋊ C₂²
derived	C₃²×H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₃×#1113
Fitting	#10624930	C₂²
socle	C₃	(C₃⋊#1113)⋊C₃
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁

533 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 27

Polycyclic presentation:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243911, #72243949, #72244021
maximal normal subgroup of	#72243949, #72244021

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