#72254704

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃×SD₁₆×3₋¹⁺² ⋊ C₂
Sylow	C₃×3₋¹⁺² ⋊ AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₄×C₃⋊AGL(1,ℤ/9)
commutator, cosocle	C₃×C₄×C₉	C₂³×C₃
Frattini	C₃×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃×SD₁₆×3₋¹⁺²	C₂
socle	C₂×C₃²	C₃×D₃×D₄
hypercenter	SD₁₆	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

770 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 216

Polycyclic presentation:

2^3,3,2^2,3^3|c^2=e,e^2=f,g^3=i,c^a=ce,c^b=cf,e^a=ef,g^b=g^2i^2,g^d=gi,h^b=h^2,i^b=i^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal normal subgroup of	AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal quotient of	#72254708, #72254769

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