Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#341 ⋊ C2 |
| Sylow | C3 ⋊ #1876 |
324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(8,10)(9,14)(11,19)(12,17)(13,21)(15,23)(16,20)(18,22);(5,7)(8,9,11,18,12,15,20,13)(10,14,19,22,17,23,16,21);(2,3)(4,5,6,7)(9,13,15,18)(10,16,17,19)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,d^2=g,e^2=g,b^a=bd,c^a=ce,h^a=h^2,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1876, C2×#68946, #10796475, … |
| maximal normal subgroup of | D3×#1876, C2×#68946, #10796475, … |
| maximal quotient of | #10689749, #10689756, #10689763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1522);