Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#575 ⋊ C2 |
| Sylow | C3 ⋊ #3019 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#937 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#575 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #575 | D3 |
| hypocenter | C3 | #3019 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #575 | |
762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,10)(9,18)(11,15)(12,19);(2,3)(7,17)(11,15)(12,19)(13,16);(4,6)(5,9)(7,16)(8,14)(10,18)(11,19)(12,15)(13,17);(4,5)(6,12)(7,15)(8,10)(9,13)(11,17)(14,19)(16,18)Polycyclic presentation:
2^7,3|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10826475, #10826476, #10826479, … |
| maximal normal subgroup of | #10826475, #10826476, #10826479, … |
| maximal quotient of | #10767555, #10767562, #10767594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16581);