Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#4565 |
| Sylow | C3 ⋊ C2×#362 |
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 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) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(3,4,5)(6,8,12,21)(7,13,16,30)(9,20,11,18)(10,24,26,28)(14,31,27,29)(15,32,33,34)(17,23,35,37)(19,25,36,22);(4,5)(6,7,9,14,12,16,11,27)(8,17,18,34,21,35,20,32)(10,23,22,33,26,37,25,15)(13,28,29,19,30,24,31,36);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4565, Dic3×#362, #10710464, … |
| maximal normal subgroup of | C4×#4565, Dic3×#362, #10710464, … |
| maximal quotient of | C2×#67764, #10704966, #10705197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4638);