Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| affine special semilinear | AΣL(1,𝔽64) |
| unambiguous semidirect product | #2846⋊C3 |
| Fitting, Sylow | #2846 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle | C25 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2846 | C3 |
| hypercenter | C22 | #120⋊C3 |
| hypocenter | C24 | C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C22 | |
627 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 627 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) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,3,7)(2,9,10,5,6,12)(4,8,11);(1,5,4,2)(3,9)(6,8)(7,10,11,12)Polycyclic presentation:
2^2,3,2^5|b^a=bd,e^a=eg,f^a=fh,e^c=efh,f^c=eg,g^c=gh,h^c=g28 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#120⋊C3, #36152⋊C3, C2×AΣL(1,𝔽64), … |
| maximal normal subgroup of | D4×#120⋊C3, #36152⋊C3, C2×AΣL(1,𝔽64), … |
| maximal quotient of | #14679⋊C3, #14693⋊C3, #14694⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5790);