Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#2355 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(14,17)(18,19)(21,22,23);(5,7,9,8)(6,10,11,12)(13,14,16,17)(15,19,20,18);(5,8,9,7)(6,10,11,12);(13,15,16,20)(14,18,17,19)150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16795);