Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#1710 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,7)(3,9,11,16)(4,12,13,8)(6,14,15,10)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(33,34,35);(1,3,4,10,5,11,13,14)(2,6,8,9,7,15,12,16)(18,22)(23,30)(24,29)(25,31);(2,7)(6,15)(8,12)(9,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5001);