Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#2981 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C3×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,12,14,20,21)(11,17,19,24,18,15,22,23)(25,26,27);(1,3)(2,5)(4,8)(6,7)(10,14)(11,18)(16,21)(19,22);(3,8)(5,7)(11,18)(15,17)(19,22)(23,24);(2,6)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17421);