Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2848⋊C3 |
| Fitting, Sylow | #2848 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle | C25 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2848 | C3 |
| prime radical | C2×C22⋊A4 | C22 |
| hypercenter | C4 | #120⋊C3 |
| hypocenter | C24 | C3×Q8 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 | |
218 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,3,4,8)(2,5,6,7)(11,12,16)(13,14,15);(1,2,4,6)(3,7,8,5)(9,13,16,10)(11,15,12,14)Polycyclic presentation:
2^2,3,2^5|a^2=d,b^2=d,b^a=bd,e^a=eg,f^a=fh,e^c=efgh,f^c=eh,g^c=gh,h^c=g28 conjugacy classes — view the full character table.
| maximal subgroup of | #32708⋊C3, #32710⋊C3, #32881⋊C3, … |
| maximal normal subgroup of | #32708⋊C3, #32710⋊C3, #32881⋊C3, … |
| maximal quotient of | C4⋊(#379⋊C3), #14693⋊C3, #15201⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5792);