Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×D3×Q32 |
| Fitting | C2×C3×Q32 ⋊ C2 |
| Sylow | C3 ⋊ C22×Q32 |
360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(3,4,5)(6,8,11,19)(7,12,15,29)(9,21,24,37)(10,26,22,18)(13,31,32,36)(14,33,25,28)(16,34,35,20)(17,30,27,23);(1,2)(6,7,11,15)(8,16,19,35)(9,20,24,34)(10,25,22,14)(12,27,29,17)(13,30,32,23)(18,36,26,31)(21,33,37,28);(4,5);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1416, #10715632, D3×#1420, … |
| maximal normal subgroup of | D3×#1416, #10715632, D3×#1420, … |
| maximal quotient of | C2×D3×#358, C2×D3×#366, C4×D3×Q32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14601);