#72219983

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

#6900 ⋊ #100

Subgroups

	subgroup	G/H
center	C₂	#72219982
commutator	C₂²×C₉²	C₂×C₃×C₄
Frattini	C₂×C₃²×C₄	C₃×D₃²
Fitting	C₂×C₄×#6900	C₂²
socle	C₂×C₃²	C₃×#3722
prime radical	C₂×C₄⋊#92314	C₂
cosocle	C₂×C₄×C₉²	C₂²×C₃
hypercenter	C₂×C₄	#92265⋊C₂
hypocenter	C₉²	C₃×#100
LCS	G ▷ C₂²×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

876 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 96 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:

2^2,3,2^3,3^4|a^2=d,d^2=f,e^2=f,g^3=i,h^3=j,b^a=bdef,d^b=df,e^a=ef,e^b=ef,g^a=g^2i^2,g^b=g^2i^2,g^c=gi^2,h^a=g^2h,h^b=h^2j^2,h^c=hj^2,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220028, #72220045, #72220054
maximal normal subgroup of	#72220028, #72220045, #72220054

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