Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C16 | C22 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C8 | C24×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ G | |
368 conjugacy classes of subgroups.
| 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);(6,7,9,13,10,14,19,25,11,15,20,26,21,27,31,35)(8,12,17,23,18,24,30,34,16,22,28,32,29,33,36,37);(8,16)(12,22)(17,28)(18,29)(23,32)(24,33)(30,36)(34,37);(1,2)(6,8)(7,12)(9,17)(10,18)(11,16)(13,23)(14,24)(15,22)(19,30)(20,28)(21,29)(25,34)(26,32)(27,33)(31,36)(35,37)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17846);