#72220612

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃³ ⋊ C₂×D₃
derived	C₃×C₃²⋊(C₂²⋊A₄) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₃²⋊(C₂²⋊A₄)	C₂²
Frattini	C₃	#13181837
Fitting	C₂⁴×C₃³	C₂×D₃
socle	C₂⁴×C₃²	D₃²
LCS	G ▷ C₃×C₃²⋊(C₂²⋊A₄)
UCS	C₁

3870 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 3870 conjugacy classes of subgroups — too many to draw. Only its 55 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=d^2,d^c=df,e^b=e^2,f^b=f^2,g^a=h,g^d=h,h^a=g,h^d=gh,i^a=gj,i^d=ij,j^a=hi,j^d=i

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72222920

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