Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C23 |
| commutator | C22 | C2×C3×C42 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C42 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ 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:
(1,2,4,8,5,7,11,14)(3,6,10,12,9,13,15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(33,34,35);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(2,7)(3,9)(8,14)(10,15)(18,23)(22,28)(24,30)(29,32)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4745);