#72220374

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×H(5,𝔽₂) ⋊ C₂×D₃
derived	C₃²×#997 ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#13181837
commutator, cosocle, hypocenter	C₃²×#997	C₂²
Fitting	C₃²×H(5,𝔽₂)	C₂×D₃
socle	C₂×C₃²	C₂×#1020
LCS	G ▷ C₃²×#997
UCS	C₁ ◁ C₂

1619 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1619 conjugacy classes of subgroups — too many to draw. Only its 41 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:

2^2,3^3,2^5|f^2=j,g^2=j,c^b=c^2,d^a=d^2,d^b=d^2,e^a=e^2,f^b=gj,f^c=fg,g^b=fj,g^c=f,g^f=gj,h^b=ghij,h^c=i,h^f=hj,h^g=hj,i^b=fgij,i^c=hi,i^f=ij

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72220377

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