#72228876

order	4563
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃² ⋊ 3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₁₃⋊C₃²
commutator, Fitting, socle, cosocle	C₃×C₁₃²	C₃²
prime radical	C₃×#92788	C₃
hypocenter	C₁₃²	3₋¹⁺²
LCS	G ▷ C₃×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₃

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

3^3,13^2|a^3=c,b^a=bc,d^a=d^3,d^b=d^9,e^a=d^2e,e^b=e^9

Conjugacy classes & character table

131 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228894, #72228912
maximal normal subgroup of	#72228894, #72228912

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [3, 3, 3, 13, 13]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^9);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);