Finite Group Databasebeta

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#72243728

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14555436 ⋊ C22

Subgroups

subgroupG/H
center, socleC3C3×C3⋊D3×(C3⋊D3)⋊C3
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC32C32×D3×C3⋊D3
Fitting#14555436C22
hypercenterC32C3⋊D3×(C3⋊D3)⋊C3
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3 ◁ C32

3462 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 3462 conjugacy classes of subgroups — too many to draw. Only its 147 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 — 3 generators on 81 points:

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

Polycyclic presentation:

2^2,3^7|d^c=di^2,e^a=e^2,e^b=e^2,e^d=eh,f^a=e^2f,f^b=f^2,f^d=fh^2i^2,g^a=gh^2i,g^b=g^2,g^e=gi,h^a=h^2i^2,h^b=h^2i^2,h^e=hi^2,h^f=hi

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243756, #72244064
maximal normal subgroup of#72243756, #72244064

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