Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1916 |
240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 240 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) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(5,7)(16,18,19,22)(17,21,20,23);(2,3)(8,10,11,15,12,9,13,14)(17,20)(21,23);(4,5)(6,7)(8,9,11,14,12,10,13,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^b=h^2,h^c=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1916, C2×#68610, #10797241, … |
| maximal normal subgroup of | D3×#1916, C2×#68610, #10797241, … |
| maximal quotient of | #10693303, #10693310, #10693317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1186);