Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | Dic3×#135 |
| Sylow | C3 ⋊ C4×#135 |
500 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 225 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 — 4 generators on 23 points:
(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,22,23);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(1,2,3,4)(22,23)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#359, #10710418, Dic3×#360, … |
| maximal normal subgroup of | Dic3×#359, #10710418, Dic3×#360, … |
| maximal quotient of | C3⋊C8×#135, #10764719, #10764937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16472);