#72243969

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552186 ⋊ C₂²
derived	C₃³×C₉ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃²
Frattini, socle	C₃³	C₃²×D₃²
Fitting	#14552186	C₂²
LCS	G ▷ C₃³×C₉
UCS	C₁

2437 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 2437 conjugacy classes of subgroups — too many to draw. Only its 99 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 — 2 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72243996

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