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×#1711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(33,34,35);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)(18,22)(23,30)(24,29)(25,31);(2,7)(6,14)(8,13)(9,15)(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, 5002);