#72220714

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃³ ⋊ C₂×D₃
derived	C₃³×C₂²⋊A₄ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₂²⋊A₄	C₂²
Frattini	C₂²	A₄⋊#1113
Fitting	C₂⁴×C₃³	C₂×D₃
socle	C₂²×C₃³	C₂×S₄
LCS	G ▷ C₃³×C₂²⋊A₄
UCS	C₁

1754 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 1754 conjugacy classes of subgroups — too many to draw. Only its 53 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:

2^2,3^4,2^4|c^a=c^2,c^b=c^2,d^a=c^2d,d^b=d^2,e^b=e^2,f^a=f^2,g^a=gij,g^b=h,g^e=h,h^a=hi,h^b=g,h^e=gh,i^b=ij,i^e=j,j^e=ij

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13166068, #72220940, #72220991
maximal normal subgroup of	D₃×#13166068, #72220940, #72220991

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.3^2*F.4);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);