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#72260092

order5832
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

FittingC22×C3×AΣL(1,𝔽27) ⋊ D3
derived#72228414 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72228414C2
FrattiniC33C3⋊(C3⋊S4)
FittingC22×C3×AΣL(1,𝔽27)D3
socleC22×C3#92338
LCSG ▷ #72228414
UCSC1

908 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 31
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 4 generators on 31 points:

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

Polycyclic presentation:

2,3^6,2^2|b^3=g^2,d^3=g,b^a=b^2g,c^a=c^2,c^b=ce,d^a=d^2g^2,d^c=def^2,e^a=eg^2,e^b=eg,f^b=fg,f^d=fg^2,g^a=g^2,h^a=hi,h^d=i,i^d=hi

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

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