Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#5070 |
| Sylow | C3 ⋊ C2×#486 |
528 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 181 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) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,8)(7,10)(9,12)(11,13)(15,18)(16,21)(19,20);(6,7,9,11)(8,13,12,10);(4,5)(8,12)(10,13)(14,15,16,19,17,20,21,18);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10690434, #10690436, #10690466, … |
| maximal normal subgroup of | #10690434, #10690436, #10690466, … |
| maximal quotient of | C2×#69574, C2×#69644, C2×#71764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15733);