#72254081

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃)×3₋¹⁺² ⋊ C₂²
derived	C₃³×C₉ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃²
Frattini	C₃²	C₃²×#1113
Fitting	C₃×H(3,𝔽₃)×3₋¹⁺²	C₂²
socle	C₃³	C₃²×D₃²
LCS	G ▷ C₃³×C₉
UCS	C₁

3449 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 3449 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 108

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2^2,3^7|f^3=i,e^a=e^2,e^c=eh,f^a=f^2i^2,f^b=f^2i^2,f^c=fi,f^d=fi,g^a=gi^2,g^b=g^2,h^a=h^2,i^a=i^2,i^b=i^2

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254082, D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal normal subgroup of	D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal quotient of	#72254082

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