#72243741

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72243615 ⋊ C₂

Subgroups

	subgroup	G/H
center, socle	C₃	C₃×H(3,𝔽₃)×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃⁴
Frattini	C₃³	C₃⁴×D₃
Fitting	#72243615	C₂
hypercenter	#6911	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₂×C₃²×H(3,𝔽₃)
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

9702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 4 generators on 81 points:

(1,12,33,4,15,13,6,3,32)(2,21,53,7,27,26,10,9,44)(5,62,36,38,14,63,19,39,34,17,37,61,16,35,40,41,18,60)(8,42,68,20,49,47,25,23,64)(11,67,56,45,28,77,29,58,54,30,46,76,22,55,59,57,31,75)(24,78,71,65,48,81,52,72,69,50,66,80,43,70,73,74,51,79);(1,11,50)(2,24,17)(3,28,70)(4,22,65)(5,30,8)(6,29,74)(7,43,38)(9,48,35)(10,52,41)(12,31,72)(13,54,79)(14,55,23)(15,46,78)(16,45,20)(18,58,42)(19,57,25)(21,51,39)(26,69,60)(27,66,62)(32,56,80)(33,59,81)(34,75,47)(36,76,64)(37,67,49)(40,77,68)(44,71,61)(53,73,63);(2,7,10)(5,16,19)(9,21,27)(11,29,22)(14,18,37)(17,38,41)(24,43,52)(26,44,53)(28,46,31)(30,57,45)(34,36,40)(35,39,62)(48,51,66)(50,65,74)(54,59,56)(55,67,58)(60,61,63)(69,71,73)(70,72,78)(75,77,76)(79,80,81);(3,12,15)(9,21,27)(13,33,32)(14,18,37)(23,42,49)(26,53,44)(28,31,46)(34,40,36)(35,39,62)(47,68,64)(48,51,66)(54,59,56)(55,58,67)(60,63,61)(69,73,71)(70,72,78)(75,77,76)(79,81,80)

Polycyclic presentation:

2,3^8|e^3=i,d^c=df,e^b=ei,f^d=fi^2,g^a=g^2,g^c=gh,g^e=gh,h^a=h^2i,h^g=hi

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243840, #72243883, #72243956
maximal normal subgroup of	#72243840, #72243883, #72243956

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, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);