Finite Group Databasebeta

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#72243738

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554864 ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC3(C3⋊D3)⋊C32
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC33C32×D32
Fitting#14554864C22
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3

1066 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 81 points:

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

Polycyclic presentation:

2^2,3^7|d^c=di,e^a=e^2,e^b=e^2,e^c=eg,e^d=eg^2,f^a=e^2f,f^b=f^2,f^c=fg^2i,f^d=fg^2h,g^a=g^2i,g^b=g^2i,g^e=gi,g^f=gi^2,h^a=g^2hi^2,h^b=h^2i^2,h^e=hi^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243756
maximal normal subgroup of#72243756

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