#72222970

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

derived

C₃×C₂²⋊H(3,𝔽₃) ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#13049215
commutator, hypocenter	C₃×C₂²⋊H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃	D₃×C₃⋊S₄
Fitting	C₂³×C₃³	C₂×D₃
socle	C₂³×C₃²	D₃²
prime radical	C₂×C₃⋊#6258	C₂
cosocle	C₂×C₃×C₂²⋊H(3,𝔽₃)	C₂²
LCS	G ▷ C₃×C₂²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂

726 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^4,2^2|b^2=c,d^a=d^2,e^a=e^2,e^b=e^2,e^d=eg,f^a=f^2,f^b=f^2,g^b=g^2,h^e=i,i^a=hi,i^b=hi,i^e=hi

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222973, #72222983, #72222986
maximal normal subgroup of	#72222973, #72222983, #72222986

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);