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#72243647

order4374
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C34

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554864 ⋊ C2

Subgroups

subgroupG/H
center, socleC3H(3,𝔽3)×(C3⋊D3)⋊C3
commutator, cosocleC3×H(3,𝔽3)C2×C33
FrattiniC33C33×D3
Fitting#14554864C2
hypercenterAΣL(1,𝔽27)(C3⋊D3)⋊C3
hypocenterH(3,𝔽3)C2×C3×H(3,𝔽3)
LCSG ▷ C3×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3 ◁ C32 ◁ AΣL(1,𝔽27)

913 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 913 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2,3^7|c^b=ch,d^c=de,e^d=eh^2,f^a=f^2,f^b=fg,f^c=fg^2,g^a=g^2h,g^f=gh

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243652, #72243738
maximal normal subgroup of#72243652, #72243738

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