#72243268

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14552544 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#13091521
commutator, cosocle	C₃×C₉²	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	#14552544	C₂
prime radical	C₃×#92299	C₃
hypocenter	C₉²	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃

682 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 162

Polycyclic presentation:

2,3^7|b^3=d,e^3=g,f^3=h,b^a=bg^2,c^b=cd^2,e^a=e^2g^2,e^c=eg,f^a=f^2h^2,f^b=fh,g^a=g^2,h^a=h^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243278, #72243339
maximal normal subgroup of	#72243278, #72243339

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