#72259581

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃³×3₋¹⁺² ⋊ C₂³
derived	C₃³×C₉ ⋊ C₂³×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂³×C₃
Frattini	C₃	C₃×#10620215
Fitting	C₃³×3₋¹⁺²	C₂³
socle	C₃⁴	C₂²×C₃×D₃
LCS	G ▷ C₃³×C₉
UCS	C₁

2335 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 2335 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 36

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

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

Polycyclic presentation:

2^3,3^6|e^3=i,e^a=e^2i^2,e^c=e^2i^2,e^d=ei^2,f^a=f^2,f^b=f^2,g^a=f^2g,g^b=g^2,h^a=hi^2,h^c=h^2,i^a=i^2,i^c=i^2

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259587
maximal quotient of	#72259587

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