Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
No aliases for this group.
| Fitting, Sylow | #14555438 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C33×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C34 |
| Frattini | C32 | C34×D3 |
| Fitting | #14555438 | C2 |
| hypercenter | C3×H(3,𝔽3) | (C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
4372 conjugacy classes of subgroups; only the 665 normal subgroups are shown.
This group has 4372 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:
(2,9,11)(4,16)(7,21,23)(8,24,27)(10,56,31,29,28,59)(14,35)(17,38)(18,39)(19,41)(20,44,42)(22,70,49,47,46,72)(25,50,51)(26,57,55,53,30,76)(34,60)(36,61)(37,62)(40,63)(43,65,64)(45,79,48,68,67,71)(52,75,58,74,54,77)(66,81,69,78,73,80);(1,17,60)(2,30,74)(3,36,39)(4,62,32)(5,14,63)(6,37,61)(7,48,78)(8,54,56)(9,55,75)(10,76,50)(11,26,77)(12,40,41)(13,19,38)(15,34,16)(18,35,33)(20,69,72)(21,45,81)(22,79,64)(23,67,80)(24,58,59)(25,31,57)(27,52,29)(28,53,51)(42,73,70)(43,46,71)(44,66,47)(49,68,65);(1,2,7)(3,8,20)(4,10,22)(5,11,21)(6,9,23)(12,24,42)(13,25,43)(14,26,45)(15,27,44)(16,29,47)(17,30,48)(18,28,49)(19,31,46)(32,50,64)(33,51,65)(34,52,66)(35,53,68)(36,54,69)(37,55,67)(38,57,71)(39,56,72)(40,58,73)(41,59,70)(60,74,78)(61,75,80)(62,76,79)(63,77,81);(3,12,15)(8,24,27)(13,33,32)(14,17,37)(20,42,44)(25,51,50)(26,30,55)(34,40,36)(35,38,62)(43,65,64)(45,48,67)(52,58,54)(53,57,76)(60,63,61)(66,73,69)(68,71,79)(74,77,75)(78,81,80)Polycyclic presentation:
2,3^7|d^3=h,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 | #72243741, #72243831, #72243870 |
| maximal normal subgroup of | #72243741, #72243831, #72243870 |
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, 4, F.8);;
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);