Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1971 |
304 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,15,16,23,17,24,26,14)(13,19,20,18,21,22,25,27);(2,3)(4,6,7,5)(9,11)(12,14,16,15,17,23,26,24)(13,18,20,22,21,27,25,19);(4,5,7,6)(8,9,10,11)(12,13)(14,22)(15,18)(16,25)(17,21)(19,23)(20,26)(24,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=e,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10712340, #10712343, #10712431, … |
| maximal normal subgroup of | #10712340, #10712343, #10712431, … |
| maximal quotient of | #10694053, #10694087, #10694351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3151);