Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1857 |
272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 272 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,12,17)(9,14,15,23)(11,20,18,16)(13,21,19,22);(4,6,7,5)(8,9)(10,14)(11,19)(12,15)(13,18)(16,21)(17,23)(20,22);(2,3)(4,5,7,6)(9,13)(11,18)(14,22)(15,19)(16,20)(21,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=g,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 | #10712009, #10712010, #10712014, … |
| maximal normal subgroup of | #10712009, #10712010, #10712014, … |
| maximal quotient of | #10689054, #10689064, #10689174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4234);