Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#424 ⋊ C2 |
| Sylow | C3 ⋊ #1866 |
270 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 270 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)(8,11)(9,14)(10,19)(12,16)(13,17)(15,22)(18,20)(21,23);(2,3)(5,7)(8,10,13,20,14,22,12,21)(9,15,16,23,11,19,17,18);(4,5)(6,7)(8,9,13,16,14,11,12,17)(10,18,20,15,22,23,21,19)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eg,h^b=h^2,d^c=dg,e^c=eg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1866, C2×#70545, #10796241, … |
| maximal normal subgroup of | D3×#1866, C2×#70545, #10796241, … |
| maximal quotient of | #10689097, #10689104, #10689111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3121);