Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #2856⋊C3 |
| Fitting, Sylow | #2856 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#127⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2856 | C3 |
| hypercenter | C22 | #127⋊C3 |
| hypocenter | C42 | C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C22 | |
194 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 194 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:
(2,4)(5,6,9)(7,13,15)(8,10,14)(11,16,12);(1,2)(3,4)(5,12)(6,13)(7,8)(9,14)(10,11)Polycyclic presentation:
2^2,3,2^5|e^2=g,f^2=h,b^a=bd,c^a=cg,e^a=eg,f^a=fh,e^c=efg,f^c=eg,g^c=gh,h^c=g28 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#127⋊C3, #36179⋊C3, C2×#2856⋊C3, … |
| maximal normal subgroup of | D4×#127⋊C3, #36179⋊C3, C2×#2856⋊C3, … |
| maximal quotient of | #14686⋊C3, #14712⋊C3, #14713⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5794);