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#72243642

order4374
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C33

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554871 ⋊ C2

Subgroups

subgroupG/H
center, socle, hypercenterC3#13091503
commutator, cosocle, hypocenterH(5,𝔽3)C2×C32
FrattiniC33C32×C3⋊D3
Fitting#14554871C2
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3

886 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2,3^7|b^3=h,c^3=h,c^b=ch,d^a=d^2,d^b=df,d^c=df^2,e^a=e^2,e^b=ef^2h,e^c=ef^2g,f^a=f^2h,f^d=fh,f^e=fh^2,g^a=g^2h^2,g^d=gh^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243644, #72243838
maximal normal subgroup of#72243644, #72243838

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