#72221985

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×#10624934 ⋊ C₂²
derived	C₃×#6908 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×#6908	C₂²×C₃
Frattini	C₃²	C₃×C₃⋊#1113
Fitting	C₃×#10624934	C₂²
socle	C₃²	C₃×((C₃⋊D₃)⋊C₃)⋊D₃
LCS	G ▷ C₃×#6908
UCS	C₁

3205 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 3205 conjugacy classes of subgroups — too many to draw. Only its 88 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:

2^2,3^7|e^3=i,d^a=d^2,e^a=e^2i^2,e^c=ei,f^b=f^2,f^d=fh,f^e=fh^2,g^a=g^2,g^b=g^2,g^f=gi^2,h^a=h^2,h^b=h^2i,h^f=hi,i^a=i^2

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222003, #72243793, D₃×#72221983
maximal normal subgroup of	D₃×#72221983

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