#72268947

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×C₄×3₋¹⁺² ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×A₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉×A₄	C₂²×C₃
Frattini	C₂×C₃	C₂×C₃×C₃⋊S₄
Fitting	C₂²×C₄×3₋¹⁺²	D₃
socle	C₂³×C₃	C₂×C₃×C₃⋊D₃
hypercenter	C₄	A₄⋊AGL(1,ℤ/9)
hypocenter	C₉×A₄	C₃×D₄
LCS	G ▷ C₂×C₉×A₄ ▷ C₉×A₄
UCS	C₁ ◁ C₂ ◁ C₄

525 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 17 points:

(1,4,3,2)(6,8)(10,16,17,12,13,14)(11,15);(5,6,7,8)(9,17,16,11,10,12,15,13,14);(2,3,4)

Polycyclic presentation:

2^2,3,2,3^3,2^2|a^2=d,f^3=g,b^a=bd,e^b=e^2,f^b=f^2g^2,f^c=fg,g^b=g^2,h^b=hi,h^e=i,i^e=hi

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×A₄⋊AGL(1,ℤ/9)
maximal normal subgroup of	D₄×A₄⋊AGL(1,ℤ/9)

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