#72220997

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃³ ⋊ C₂²×D₃
derived	C₃³×A₄ ⋊ C₂³

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×A₄	C₂³
Fitting, socle	C₂²×C₃³	C₂²×D₃
LCS	G ▷ C₃³×A₄
UCS	C₁

1568 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 68 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|d^a=d^2,d^c=d^2,e^a=d^2e,e^c=e^2,f^b=f^2,g^a=g^2,g^b=g^2,h^b=hi,h^f=i,i^f=hi

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222935, #72222979, #72222994, …
maximal normal subgroup of	D₃³×S₄
maximal quotient of	#72222935, #72222979

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