#72220028

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×#6900 ⋊ C₂²
Sylow	#6900 ⋊ #453

Subgroups

	subgroup	G/H
center	C₂	#72220027
commutator, cosocle	C₂×C₄×C₉²	C₂³×C₃
Frattini	C₂×C₃²×C₄	C₂×C₃×D₃²
Fitting	C₄²×#6900	C₂²
socle	C₂×C₃²	C₃×#63923
hypercenter	C₄²	#92265⋊C₂
hypocenter	C₉²	C₃×#453
LCS	G ▷ C₂×C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄²

2445 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 2445 conjugacy classes of subgroups — too many to draw. Only its 161 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:

2^3,3,2^3,3^4|a^2=g,b^2=e,f^2=g,h^3=j,i^3=k,b^a=bf,c^b=ce,e^a=eg,f^a=fg,f^c=fg,h^a=h^2j^2,h^c=h^2j^2,h^d=hj^2,i^a=h^2i,i^c=i^2k^2,i^d=ik^2,j^a=j^2,j^c=j^2,k^a=j^2k,k^c=k^2

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220092
maximal normal subgroup of	#72220092

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^2*F.11^2);;
gap> SetConjugate(coll, 9, 4, F.9*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 3, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);