#72243277

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14555345 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃×#4207⋊C₃
commutator, cosocle	C₃²×C₉	C₂×C₃³
Frattini, socle	C₃²	C₃³×C₃⋊D₃
Fitting	#14555345	C₂
prime radical	C₃×#4207⋊C₃	C₃
hypercenter	C₉	#4207⋊C₃
hypocenter	C₃×C₉	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃ ◁ C₉

2425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 36 points:

(1,4,9,3,7,5,6,2,8)(10,11)(12,26)(13,22)(15,17)(16,21)(18,19)(20,36)(23,25)(24,30)(27,31)(28,33)(29,34)(32,35);(2,4,7)(5,9,8)(10,16,21,11,23,22,14,13,25)(12,20,31,15,24,34,18,28,32)(17,29,33,26,27,30,19,35,36);(10,18,17,14,15,19,11,12,26)(13,20,29,23,28,35,16,24,27)(21,31,36,25,32,30,22,34,33)

Polycyclic presentation:

2,3^7|d^3=e,f^3=h,g^3=h^2,d^a=dh^2,d^c=de,f^a=f^2h^2,f^c=fh,g^a=g^2h,g^b=gh^2,h^a=h^2

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243364
maximal normal subgroup of	#72243364
maximal quotient of	#72243278, #72243916

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