Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#5195 |
| Sylow | C3 ⋊ C2×#510 |
360 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(3,4,5)(6,8)(7,12)(9,21)(10,26)(11,17)(13,31)(14,33)(15,28)(16,20)(18,23)(19,22)(24,37)(25,29)(27,30)(32,35)(34,36);(1,2)(8,17)(12,28)(16,34)(18,27)(19,26)(21,37)(29,33)(31,35);(4,5)(6,7,11,15)(8,16,17,34)(9,20,24,36)(10,25,22,14)(12,27,28,18)(13,30,32,23)(19,35,26,31)(21,33,37,29);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10712319, #10712321, #10712340, … |
| maximal normal subgroup of | #10712319, #10712321, #10712340, … |
| maximal quotient of | C2×#70654, C2×#72078, C2×#70649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16471);