Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2 |
| direct product | C3×#1308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #95 |
| commutator | C23 | C3×C42 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18,20,23)(19,22)(21,28,27,24)(25,30)(26,32)(29,31)(33,34,35);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,21,26,20,25,27,31)(18,22,24,29,23,30,28,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 444);