Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
No aliases for this group.
| Fitting, Sylow | #14555436 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C33×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C34 |
| Frattini | C32 | C34×D3 |
| Fitting | #14555436 | C2 |
| hypercenter | C3×H(3,𝔽3) | (C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
4462 conjugacy classes of subgroups; only the 665 normal subgroups are shown.
This group has 4462 conjugacy classes of subgroups — too many to draw. Only its 665 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 81 points:
(3,13,15)(4,16)(8,25,24)(10,28)(12,30,32)(14,56,35,34,33,57)(17,36)(18,37)(19,39,41)(21,43)(23,45,47)(26,70,48,50,49,69)(27,51)(29,52)(31,71,55,54,53,72)(38,59,58)(40,76,63,62,61,77)(42,64)(44,65)(46,78,68,67,66,79)(60,81,73,75,74,80);(1,4,36,5,18,16,6,17,37)(2,27,51,9,29,52,11,10,28)(3,14,56,15,35,34,13,33,57)(7,42,43,22,21,65,20,44,64)(8,48,69,24,49,70,25,26,50)(12,31,72,30,53,54,32,55,71)(19,61,62,41,40,77,39,63,76)(23,66,78,45,68,79,47,46,67)(38,73,75,59,60,81,58,74,80);(1,3,12)(2,8,23)(4,14,31)(5,15,30)(6,13,32)(7,19,38)(9,24,45)(10,26,46)(11,25,47)(16,34,54)(17,33,55)(18,35,53)(20,39,58)(21,40,60)(22,41,59)(27,48,66)(28,50,67)(29,49,68)(36,56,72)(37,57,71)(42,61,73)(43,62,75)(44,63,74)(51,69,78)(52,70,79)(64,76,80)(65,77,81);(1,2,7)(3,8,19)(4,10,21)(5,9,22)(6,11,20)(12,23,38)(13,25,39)(14,26,40)(15,24,41)(16,28,43)(17,29,42)(18,27,44)(30,45,59)(31,46,60)(32,47,58)(33,49,61)(34,50,62)(35,48,63)(36,52,64)(37,51,65)(53,66,74)(54,67,75)(55,68,73)(56,70,76)(57,69,77)(71,78,81)(72,79,80)Polycyclic presentation:
2,3^7|d^b=dh,e^c=eh^2,f^a=f^2,f^d=fg,g^a=g^2h,g^f=gh274 conjugacy classes — view the full character table.
| maximal subgroup of | #72243652, #72243728, #72243811 |
| maximal normal subgroup of | #72243652, #72243728, #72243811 |
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, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
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);