Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | C4⋊(#127⋊C3) |
| unambiguous semidirect product | #2867⋊C3 |
| Fitting, Sylow | #2867 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#127⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2867 | C3 |
| hypercenter | C4 | #127⋊C3 |
| hypocenter | C42 | C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C4 | |
252 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 252 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:
(1,6,12,2,8,11,4,9,7,5,3,10);(1,5)(2,4)(3,8)(7,11)(10,12)Polycyclic presentation:
2^2,3,2^5|b^2=d,e^2=g,f^2=h,b^a=bd,e^a=eg,f^a=fh,c^b=cgh,e^c=fh,f^c=efh,g^c=h,h^c=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #32796⋊C3, #32798⋊C3, #33160⋊C3, … |
| maximal normal subgroup of | #32796⋊C3, #32798⋊C3, #33160⋊C3, … |
| maximal quotient of | C4⋊(#384⋊C3), #14713⋊C3, C4⋊(#392⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5797);