#72220306

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ SD₁₆
Sylow	C₃⁴ ⋊ #465

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#13049804
commutator, cosocle	C₂×#66472	C₂³
Fitting	C₂²×C₃⁴	SD₁₆
socle	C₂×C₃⁴	C₂×SD₁₆
prime radical	#72219709	C₂
hypercenter	C₂²	#13049804
hypocenter	C₃⁴	#465
LCS	G ▷ C₂×#66472 ▷ C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1071 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 38 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=d,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,g^c=h,g^d=gh,g^f=g^2,h^a=g^2h,h^c=gh,h^d=gh^2,h^f=h^2,i^a=g^2h^2i^2,i^c=g^2j,i^d=ij^2,i^f=i^2,j^a=h^2ij,j^c=hij^2,j^d=i^2j^2,j^f=j^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220308, #72222921
maximal normal subgroup of	#72220308, #72222921

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