Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#907 |
| Fitting | C22×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ C22×#131 |
1504 conjugacy classes of subgroups; only the 821 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 821 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) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(5,6,7)(8,10,11,17,12,16,18,22)(9,14,15,21,13,19,20,23);(1,2)(3,4)(9,13)(10,16)(15,20)(17,22);(6,7)(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23);(3,4);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10700600, #10700604, #10700618, … |
| maximal normal subgroup of | #10700600, #10700604, #10700618, … |
| maximal quotient of | C22×#4708, C22×#5122, C2×C4×#907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19729);