Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#2337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,3)(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,22,23);(1,3,4,2)(6,9)(7,12);(5,7,8,12)(6,10,9,11)(14,17)(18,20);(6,9)(10,11)(13,15,16,19)(14,18,17,20)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16777);