#72277879

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₃³×H(3,𝔽₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	#14519546
commutator, Fitting, cosocle, hypocenter	C₃³×H(3,𝔽₃)	C₂³
socle	C₃⁴	C₂²×C₃⋊D₃
LCS	G ▷ C₃³×H(3,𝔽₃)
UCS	C₁ ◁ C₃

5053 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 5053 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 18
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(5,6)(7,8,9)(11,14,15)(13,17,18);(4,5,6)(8,9)(10,11)(12,15)(14,16);(2,3)(11,13)(14,18)(15,17)

Polycyclic presentation:

2^3,3^6|d^a=d^2,d^c=d^2,e^a=e^2,e^b=e^2,f^a=f^2,f^b=f^2,f^e=fi,g^c=g^2,h^b=h^2

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72277883

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