#72243293

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552587 ⋊ C₂²
derived	C₉³ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉³	C₂²×C₃
Frattini, socle	C₃³	C₃×D₃×C₃⋊D₃
Fitting	#14552587	C₂²
LCS	G ▷ C₉³
UCS	C₁

1680 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 27
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 27 points:

(2,8,9,4,5,6)(3,7)(11,17,18,13,14,15)(12,16)(19,23,24,21,27,26,25,20,22);(10,14,13,12,18,15,16,11,17)(20,22)(21,25)(23,26)(24,27);(1,5,4,3,9,6,7,2,8)

Polycyclic presentation:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#92314, #72243365, #72243389
maximal normal subgroup of	AGL(1,ℤ/9)×#92314, #72243365, #72243389

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