#72222985

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×((C₃⋊S₄)⋊C₃)⋊C₂
commutator, cosocle	C₂×C₃×C₂²⋊H(3,𝔽₃)	C₂³
Frattini	C₂×C₃	D₃²×S₄
Fitting	C₂³×C₃³	C₂²×D₃
socle	C₂³×C₃²	C₂×D₃²
hypocenter	C₃×C₂²⋊H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×C₃×C₂²⋊H(3,𝔽₃) ▷ C₃×C₂²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂

2474 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 2474 conjugacy classes of subgroups — too many to draw. Only its 76 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:

2^4,3^4,2^2|b^a=bd,e^b=e^2,e^c=e^2,f^a=f^2,f^b=f^2,f^e=fh,g^a=gh^2,g^c=g^2,h^a=h^2,h^c=h^2,i^f=j,j^a=ij,j^b=ij,j^f=ij

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72222993

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