#72277776

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552185 ⋊ C₂
derived	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
commutator, cosocle, hypocenter	C₃×C₃²⋊H(3,𝔽₃)	C₂×C₃
Frattini, socle	C₃³	C₃×C₃⋊(C₃⋊D₃)
Fitting	#14552185	C₂
LCS	G ▷ C₃×C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃²

4921 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 4921 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 27 points:

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

Polycyclic presentation:

2,3^7|c^a=c^2,c^b=cf,d^a=d^2,d^c=dh,e^a=e^2,e^d=eg,f^a=f^2

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72277778

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