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#72243661

order4374
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C34

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#14554861 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#14554861C2
FrattiniC33C3⋊(C3⋊(C3⋊D3))
socleC3#13091502
LCSG ▷ #14554861
UCSC1

1669 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1669 conjugacy classes of subgroups — too many to draw. Only its 229 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 — 5 generators on 81 points:

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

Polycyclic presentation:

2,3^7|e^3=h,b^a=b^2,c^a=c^2,c^b=cf,d^a=d^2,d^b=dg,e^a=e^2h^2,e^c=ef^2,e^d=efh^2,f^a=fh^2,f^c=fh,f^d=fh^2,g^c=gh^2,h^a=h^2

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243670, #72243772, #72243868
maximal normal subgroup of#72243670, #72243772, #72243868

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