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#72243786

order8748
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C32

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#14554857 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#14554857C22
FrattiniC33D3×C3⋊(C3⋊D3)
socleC3#72214354
LCSG ▷ #14554857
UCSC1

1620 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 81

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

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

Polycyclic presentation:

2^2,3^7|f^3=i^2,c^b=c^2,d^a=d^2,d^b=d^2,d^c=dg,e^a=d^2e,e^b=e^2,e^c=eh,f^b=f^2i,f^d=fg^2,f^e=fgi^2,g^a=g^2i,g^b=gi^2,g^d=gi,g^e=gi^2,h^a=g^2hi^2,h^d=hi^2,i^b=i^2

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243866, #72244026
maximal normal subgroup of#72243866, #72244026

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