Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#2104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #401 |
| commutator, Frattini, cosocle | C24 | C23×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(3,9,11,6)(4,7,13,16)(5,8)(10,15)(12,14)(17,18,19);(1,3)(2,6)(4,12,13,10)(5,11)(7,15,16,14)(8,9);(1,4)(2,7,8,16)(3,10)(5,13)(6,14,9,15)(11,12)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5395);