#72243748

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552809 ⋊ C₂
derived	C₃×C₉² ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉²	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	#14552809	C₂
LCS	G ▷ C₃×C₉²
UCS	C₁

1144 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 243

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

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

Polycyclic presentation:

2,3^7|d^3=h,e^3=g,d^a=d^2h^2,d^b=df,d^c=dh,e^a=e^2g^2,e^b=eg^2h^2,e^c=eg,f^a=f^2,g^a=g^2,h^a=h^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72243764

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