Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#4548 |
| Sylow | C3 ⋊ C2×#356 |
240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(3,4,5)(6,8,11,9)(7,12,15,20)(10,18,19,17)(13,21,16,14);(4,5)(6,7,9,13,11,15,8,16)(10,14,17,20,19,21,18,12);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682920, #10682921, #10682944, … |
| maximal normal subgroup of | #10682920, #10682921, #10682944, … |
| maximal quotient of | C2×#67643, C2×#67635, #10704469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4372);