#72243458

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃)×3₋¹⁺² ⋊ C₂
derived	C₃×C₉×H(3,𝔽₃) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₃⋊(C₃⋊AGL(1,ℤ/9)))
commutator, cosocle, hypocenter	C₃×C₉×H(3,𝔽₃)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊(C₃⋊(C₃⋊D₃))
Fitting	C₃×H(3,𝔽₃)×3₋¹⁺²	C₂
socle	C₃³	C₃×C₃⋊(C₃⋊D₃)
LCS	G ▷ C₃×C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₃

6211 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 6211 conjugacy classes of subgroups — too many to draw. Only its 638 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) = 21
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253773, #72253826
maximal normal subgroup of	#72253773, #72253826
maximal quotient of	#72243459

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