Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 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);(7,12)(8,16)(9,19)(10,21)(13,27)(14,30)(15,26)(17,33)(20,23)(22,24)(25,29)(28,34)(31,35)(36,37);(6,7,11,15)(8,13,16,29)(9,20,22,30)(10,23,21,14)(12,24,26,19)(17,34,31,37)(18,36,32,28)(25,33,27,35);(1,2)(6,8)(7,13)(9,17)(10,18)(11,16)(12,25)(14,28)(15,29)(19,35)(20,34)(21,32)(22,31)(23,36)(24,33)(26,27)(30,37)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17853);