Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#5022 |
| Sylow | C3 ⋊ C2×#479 |
384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 384 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) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(3,4,5)(14,16,18,24)(15,20,19,28)(17,26,27,23)(21,29,25,22);(1,2)(6,8,9,13)(7,10,11,12)(15,19)(20,28)(21,25)(22,29);(4,5)(6,7,9,11)(8,12,13,10)(14,15,18,19)(16,22,24,29)(17,25,27,21)(20,26,28,23);(1,2)78 conjugacy classes — view the full character table.
| maximal subgroup of | #10691541, #10691543, #10802801, … |
| maximal normal subgroup of | #10691541, #10691543, #10802801, … |
| maximal quotient of | C2×#71627, C2×#71628, C2×#71629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15006);