Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#5204 |
| Sylow | C3 ⋊ C2×#473 |
448 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 141 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:
(1,2)(3,4,5)(6,8,11,20)(7,13,16,29)(9,22,24,19)(10,26,25,18)(12,27,15,32)(14,30,21,17)(23,35,37,36)(28,34,31,33);(1,2)(7,12)(8,18)(10,25)(13,29)(14,28)(15,16)(17,30)(19,35)(20,26)(21,31)(22,36)(23,37);(4,5)(6,7,11,16)(8,17,20,30)(9,21,24,14)(10,12,25,15)(13,22,29,19)(18,34,26,33)(23,31,37,28)(27,35,32,36);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5204, #10802075, #10802096, … |
| maximal normal subgroup of | C4×#5204, #10802075, #10802096, … |
| maximal quotient of | C2×#71101, C2×#71037, C2×#71102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16553);