#72253771

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14555275 ⋊ C₂
derived	C₉×H(3,𝔽₃) ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(#4207⋊C₃)
commutator, cosocle, hypocenter	C₉×H(3,𝔽₃)	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊(C₃⋊D₃)
Fitting	#14555275	C₂
LCS	G ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₃

4330 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 4330 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 36
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,9)(6,8)(11,23,17,14,15,25)(12,21,19,18,20,30)(13,22)(16,35,26,27,28,33)(24,31)(29,34)(32,36);(10,19,21,13,20,30,22,12,18)(11,28,29,17,26,36,15,16,31)(14,32,33,23,34,35,25,24,27);(2,5,6)(4,8,9);(1,2,4)(3,6,8)(5,9,7)(10,11,14)(12,16,24)(13,17,23)(15,25,22)(18,31,27)(19,28,32)(20,26,34)(21,29,33)(30,36,35)

Polycyclic presentation:

2,3^7|f^3=g,d^a=d^2,d^c=dg^2,e^a=e^2,e^d=eh,f^a=f^2g^2,f^b=fg,f^c=fg^2,g^a=g^2

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72253773

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