Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | D3×#519 |
| Fitting | C3×#519 ⋊ C2 |
| Sylow | C3 ⋊ C2×#519 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting | C3×#519 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C23×D3 | C23 |
| hypercenter | #519 | D3 |
| hypocenter | C3 | C2×#519 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #519 | |
1488 conjugacy classes of subgroups; only the 813 normal subgroups are shown.
This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 813 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 — 5 generators on 23 points:
(6,12)(10,13)(11,20)(18,19)(21,22,23);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(22,23)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1830, D3×#1857, #10796012, … |
| maximal normal subgroup of | D3×#1830, D3×#1857, #10796012, … |
| maximal quotient of | D3×#2274, D3×#2276, #10953434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18456);