Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#5129 |
| Fitting | C22×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#408 |
792 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 269 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:
(3,4,5)(7,9)(10,12,13,19,14,20,21,25)(11,16,17,23,15,18,22,24);(1,2)(10,11)(12,18)(13,17)(14,15)(16,20)(19,24)(21,22)(23,25);(4,5)(6,7)(8,9)(11,15)(16,18)(17,22)(23,24);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10700538, #10700543, #10700563, … |
| maximal normal subgroup of | #10700538, #10700543, #10700563, … |
| maximal quotient of | C2×#69969, C2×#70828, C2×#70831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16337);