Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C3×C42 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C22×C3 | #107 |
| prime radical | C23×C3 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 39 points:
(5,6,9,14,11,17,23,16)(7,18,20,32,22,33,34,27)(8,15,10,13,24,30,25,31)(12,21,26,35,29,36,28,19)(37,38,39);(1,2,3,4)(5,7,10,21,9,20,24,35,11,22,25,36,23,34,8,19)(6,12,15,27,16,28,31,33,17,29,30,32,14,26,13,18)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 517);