Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#5167 |
| Sylow | C3 ⋊ C2×#497 |
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5)(7,9)(10,12)(11,15)(13,22)(14,18)(16,25)(17,24)(19,21)(20,23);(1,2)(7,9)(10,11,14,17)(12,19,18,25)(13,21,23,16)(15,22,24,20);(4,5)(6,7,8,9)(12,18)(15,24)(19,25)(20,22);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5167, #10803739, #10803757, … |
| maximal normal subgroup of | C4×#5167, #10803739, #10803757, … |
| maximal quotient of | C2×#71258, C2×#71257, C2×#71337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16429);