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 ⋊ #1993 |
276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,11,15)(9,12,13,14)(17,21)(18,23);(4,6,7,5)(16,18,22,17,20,23,19,21);(2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17)(18,22)(19,23)(20,21)Polycyclic presentation:
2^7,3|a^2=f,b^2=de,c^2=efg,d^2=g,b^a=bd,c^a=cef,d^a=dg,h^a=h^2,c^b=cg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10687583, #10687585, #10690100, … |
| maximal normal subgroup of | #10687583, #10687585, #10690100, … |
| maximal quotient of | #10695206, #10695222, #10695270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2404);