Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3400 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ G | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,6)(7,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)(25,26,27);(1,2,5,6)(3,7,4,8)(11,18)(14,23)(17,24)(19,21);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(1,4,5,3)(2,7,6,8)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17840);