#72220759

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₄⋊C₄
Sylow	C₃⁴ ⋊ #385

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#13049783
commutator	C₂×C₃²×C₃⋊D₃	C₂²×C₄
Fitting	C₂²×C₃⁴	C₄⋊C₄
socle	C₂×C₃⁴	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃²	C₂²
cosocle	C₂×C₃⋊D₃²	C₂³
hypercenter	C₂²	#13049783
hypocenter	C₃⁴	#385
LCS	G ▷ C₂×C₃²×C₃⋊D₃ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1598 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1598 conjugacy classes of subgroups — too many to draw. Only its 75 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:

2^6,3^4|a^2=e,c^2=f,b^a=bd,c^a=ce,g^a=gh,g^c=gh^2,g^e=g^2,g^f=g^2,h^a=gh^2,h^c=g^2h^2,h^e=h^2,h^f=h^2,i^a=g^2hi,i^c=ij,i^e=g^2h^2i,i^f=i^2,j^a=hj,j^c=ij^2,j^e=gj,j^f=j^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220785
maximal normal subgroup of	#72220785

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