#72220549

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₂×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#13049813
commutator	C₃⋊(C₃⋊(C₃⋊D₃))	C₂²×C₄
Fitting, socle	C₂×C₃⁴	C₂×D₄
prime radical	C₂×C₃⋊D₃²	C₂²
cosocle	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂³
hypocenter	C₃⁴	C₂×C₄⋊C₄
LCS	G ▷ C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁ ◁ C₂

796 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 50 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:

2^5,3^4|a^2=d,b^2=e,b^a=be,f^b=fg,f^c=f^2,f^e=f^2,g^a=fg^2,g^b=fg^2,g^c=g^2,g^e=g^2,h^a=f^2h^2,h^b=h^2i^2,h^c=f^2gh,h^e=h^2,i^a=g^2h^2i,i^b=h^2i,i^c=fgi,i^e=i^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220560, #72220566, #72220567, …
maximal normal subgroup of	#72220560, #72220566, #72220567, …
maximal quotient of	#72252786, #72269690

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