#72220772

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ #99

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊#157
commutator	C₂×C₃²×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃⁴	C₂²⋊C₄
prime radical	#13049504	C₂
cosocle	C₂×C₃⋊D₃²	C₂²
hypocenter	C₃⁴	#99
LCS	G ▷ C₂×C₃²×C₃⋊D₃ ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂

660 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 21 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=e,b^2=c,b^a=bd,c^a=ce,d^b=de,f^a=fh,f^b=fh,f^c=f^2,f^d=f^2,g^a=gh,g^b=gi,g^c=g^2,g^d=fg,h^a=h^2,h^b=fh^2,h^c=h^2,h^d=h^2,i^a=hi,i^b=gi^2,i^c=i^2,i^d=hi

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220778
maximal normal subgroup of	#72220778

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