Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#2245 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,28,18)(12,29,25,17)(13,20,32,34)(14,19,24,38)(15,33,31,21)(26,39,30,36)(27,37,40,35)(41,42,43);(1,3,4,7)(2,5,6,8)(9,11,24,30,15,27,13,25,16,28,14,26,31,40,32,12)(10,17,34,37,21,36,19,23,22,29,20,35,33,39,38,18);(2,6)(5,8)(9,12,13,27,31,26,24,11,16,25,32,40,15,30,14,28)(10,18,19,36,33,35,34,17,22,23,38,39,21,37,20,29)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5536);