Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#1526 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C23×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,6,7,8)(9,10,13,11,14,16,12,15)(17,18,19);(1,2)(3,6)(4,8)(5,7)(9,11,14,15)(10,13,16,12);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,16,11)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4817);