Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#5082 |
| Sylow | C3 ⋊ C2×#491 |
384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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:
(1,2)(3,4,5)(6,8)(7,10)(9,12)(11,13)(14,16,18,24)(15,20,19,27)(17,25,26,23)(21,29,28,22);(6,7,9,11)(8,13,12,10)(15,19)(20,27)(21,28)(22,29);(4,5)(8,12)(10,13)(14,15,17,21,18,19,26,28)(16,22,23,27,24,29,25,20);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10690128, #10696496, #10696497, … |
| maximal normal subgroup of | #10690128, #10696496, #10696497, … |
| maximal quotient of | C2×#69575, C2×#69675, C2×#71786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15773);