Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#1924 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(3,4)(6,7)(9,10,12,14,11,13,15,16)(17,18,19);(5,6,8,7)(9,10,12,14,11,13,15,16);(2,4)(5,7,8,6)(10,13)(14,16)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5215);