Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5246 |
| Sylow | C3 ⋊ C2×#456 |
708 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 708 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) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(3,4,5)(6,9,11,7)(8,15,16,12)(10,13,20,18)(14,19,21,17);(1,2)(6,8)(7,12)(9,15)(10,19)(11,16)(13,21)(14,18)(17,20);(1,2)(4,5)(6,7,11,9)(8,14,16,21)(10,18,20,13)(12,19,15,17);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | #10712531, #10712541, #10714222, … |
| maximal normal subgroup of | #10712531, #10712541, #10714222, … |
| maximal quotient of | C2×#71889, C2×#71886, C2×#71880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16694);