#72243231

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552206 ⋊ 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	#14552206	C₂
LCS	G ▷ C₃×C₉²
UCS	C₁

1678 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1678 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243235, #72243361, #72243764
maximal normal subgroup of	#72243361

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