#72243545

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552186 ⋊ C₂
derived	C₃×C₃²⋊C₉ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#13091513
commutator, cosocle, hypocenter	C₃×C₃²⋊C₉	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	#14552186	C₂
LCS	G ▷ C₃×C₃²⋊C₉
UCS	C₁ ◁ C₃

2722 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 2722 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 27
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 27 points:

(2,7,9,4,5,8)(3,6)(11,13)(12,16)(14,17)(15,18)(20,22)(23,27)(24,26);(1,2,4)(3,5,8)(6,9,7)(10,14,15,12,18,17,16,11,13)(20,23,24)(22,26,27);(1,2,4)(3,5,8)(6,9,7)(19,20,22)(21,24,26)(23,27,25)

Polycyclic presentation:

2,3^7|e^3=g,d^a=d^2,d^c=df,e^a=e^2g^2,e^b=eg,e^c=ef,e^d=eh,f^a=f^2,g^a=g^2

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253773, #72269402
maximal normal subgroup of	#72269402

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