#72222714

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#35920 ⋊ C₃²
derived	#2276 ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	A₄×#126⋊C₃
commutator, cosocle, hypocenter	#2276	C₂×C₃²
Frattini, socle	C₂³	C₂×A₄²
Fitting	#35920	C₃²
LCS	G ▷ #2276
UCS	C₁ ◁ C₂

302 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2,3^2,2^7|d^2=j,e^2=j,f^2=h,g^2=i,d^b=ej,d^c=dej,e^b=de,e^c=dj,e^d=ej,f^a=fi,f^b=fg,f^c=dfj,f^d=fj,f^e=fj,g^a=ghi,g^b=fi,g^c=deg,g^d=gj,h^b=hi,i^b=h

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228147
maximal normal subgroup of	#72228147
maximal quotient of	#72222716

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