Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #14552619 ⋊ C2 |
| derived | #10624547 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊AGL(1,ℤ/9)) |
| commutator, cosocle, hypocenter | #10624547 | C2×C3 |
| Frattini, socle | C33 | C3×C3⋊(C3⋊D3) |
| Fitting | #14552619 | C2 |
| LCS | G ▷ #10624547 | |
| UCS | C1 ◁ C32 | |
11722 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 11722 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,8,9,4,5,6)(3,7)(11,13)(14,18)(15,17)(20,22)(23,26)(25,27);(11,14,15)(13,17,18)(20,23,27)(22,25,26);(11,14,15)(13,17,18);(1,5,4,3,9,6,7,2,8)(10,11,13)(12,15,17)(14,18,16)(19,20,22)(21,23,26)(24,27,25)Polycyclic presentation:
2,3^7|c^3=f,c^a=c^2f^2,c^b=cf,d^a=d^2,d^c=dg,e^a=e^2,e^c=egh^2,f^a=f^2198 conjugacy classes — view the full character table.
| maximal subgroup of | #72243459, #72253819 |
| maximal normal subgroup of | #72253819 |
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, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);