Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1385 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #106 |
| commutator | C8 | C3×C42 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C2×C3 | #340 |
| prime radical | C23×C3 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,11,8,13,7,23)(3,14,16,29,18,30,21,25)(4,12,6,22,20,28,19,10)(9,17,24,15,27,31,26,32)(33,34,35);(1,3,6,17,5,21,20,32,8,18,19,31,7,16,4,15)(2,9,12,25,23,24,10,14,13,27,28,29,11,26,22,30)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 521);