#72222296

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14551762 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#92191⋊C₃
commutator, cosocle	C₃×#732	C₂×C₃²
Frattini	C₃²×C₉	C₃²×D₃
Fitting	#14551762	C₂
socle	C₃²	#92285
prime radical	C₃×#92333	C₃
hypocenter	#732	C₂×3₋¹⁺²
LCS	G ▷ C₃×#732 ▷ #732
UCS	C₁ ◁ C₃

452 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 33 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:

2,3^7|c^3=d,f^3=h^2,c^b=cd,e^a=e^2,e^c=ef,e^d=eh^2,f^a=f^2g,f^b=fh^2,f^e=fg,g^a=gh,g^e=gh,h^a=h^2

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72222308

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