Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| socle | C22×C3 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
368 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 368 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)(9,19)(10,21)(13,24)(14,28)(15,25)(16,29)(17,31)(20,23)(22,34)(26,36)(27,35)(30,33)(32,37);(6,7)(8,13)(9,20)(10,23)(11,15)(12,19)(14,21)(16,30)(17,33)(18,27)(22,28)(24,29)(25,34)(26,31)(32,36)(35,37);(1,2)(6,8,11,18)(7,13,15,27)(9,16,22,32)(10,17,21,31)(12,24,25,35)(14,26,23,33)(19,29,34,37)(20,30,28,36)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17851);