Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#947 |
| Fitting | C23×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ C22×#124 |
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(5,6,7)(8,9)(10,11)(12,14,15,19)(13,17,16,18);(1,2)(3,4)(12,13,15,16)(14,18,19,17);(6,7)(9,11)(13,16)(17,18);(3,4);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10687822, #10701728, #10701731, … |
| maximal normal subgroup of | #10687822, #10701728, #10701731, … |
| maximal quotient of | C22×#4977, C22×#4990, C22×#4988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19821);