Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#393 ⋊ C2 |
| Sylow | C3 ⋊ #2561 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#393 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #393 | D3 |
| hypocenter | C3 | #2561 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #393 | |
672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 672 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(9,13)(12,14);(5,7)(18,22)(20,23);(8,10,11,15)(9,12,13,14)(16,18,19,22)(17,20,21,23);(2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=eg,b^a=be,c^a=cfg,d^a=de,h^a=h^2,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10705672, #10706074, #10769297, … |
| maximal normal subgroup of | #10705672, #10706074, #10769297, … |
| maximal quotient of | #10717407, #10717946, #10717996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9444);