Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1856 |
324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(10,14)(12,15);(4,6,7,5)(8,9)(10,15)(11,13)(12,14);(2,3)(4,5,7,6)(9,12)(10,14)(13,15)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2,d^b=dg,e^b=eg,d^c=dg66 conjugacy classes — view the full character table.
| maximal subgroup of | #10711937, #10711938, #10711944, … |
| maximal normal subgroup of | #10711937, #10711938, #10711944, … |
| maximal quotient of | #10689049, #10689059, #10689163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4223);