Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#5247 |
| Sylow | C3 ⋊ C2×#379 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator | C22×C3 | C23×C4 |
| Frattini | C23 | C23×D3 |
| Fitting, prime radical | C26×C3 | C2 |
| socle | C24×C3 | C23 |
| cosocle | C23×C3 | C24 |
| hypercenter | C26 | D3 |
| hypocenter | C3 | C2×#379 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C26 | |
3464 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 3464 conjugacy classes of subgroups — too many to draw. Only its 549 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 17 points:
(3,4,5)(6,7)(8,9);(10,11)(12,13)(14,15)(16,17);(4,5)(7,9)(11,13)(14,15,16,17);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3⋊#1304, #10685511, #10700448, … |
| maximal normal subgroup of | C2×C3⋊#1304, #10685511, #10700448, … |
| maximal quotient of | C2×#72127, C4×#5247, C2×#72131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16703);