#72259402

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₉² ⋊ C₂²
Sylow	C₃×C₉² ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₉×C₃⋊D₉
commutator, cosocle	C₂×C₃×C₉²	C₂³
Frattini	C₂×C₃²	C₂×D₃×C₃⋊D₃
Fitting	C₂²×C₃×C₉²	C₂²
socle	C₂×C₃³	C₂×D₃²
hypercenter	C₂²	D₉×C₃⋊D₉
hypocenter	C₃×C₉²	C₂×D₄
LCS	G ▷ C₂×C₃×C₉² ▷ C₃×C₉²
UCS	C₁ ◁ C₂ ◁ C₂²

1578 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 108

Polycyclic presentation:

2^4,3^5|a^2=d,e^3=h,f^3=i,b^a=bd,e^a=e^2h^2,e^c=e^2h^2,f^a=e^2f,f^c=f^2i^2,g^c=g^2,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2

Conjugacy classes & character table

303 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13047160, #72259878
maximal normal subgroup of	D₃×#13047160, #72259878

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