Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#3216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
390 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)(9,10,13,17)(11,18,19,24)(12,15,22,16)(14,20,23,21)(25,26,27);(1,3,4,8)(2,5,6,7)(11,19)(14,23)(18,24)(20,21);(3,8)(5,7)(9,11)(10,14)(12,20)(13,19)(15,18)(16,24)(17,23)(21,22);(2,6)(5,7)(9,12,13,22)(10,15,17,16)(11,20,19,21)(14,18,23,24)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17656);