Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#2950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,21)(10,13)(11,17)(14,23)(16,29)(18,28)(19,27)(20,33)(22,25)(24,31)(26,32)(30,36)(34,35)(37,38,39);(5,7,10,18)(6,11,13,25)(8,16,19,30)(9,22,23,17)(12,24,27,35)(14,28,21,15)(20,34,32,31)(26,36,33,29);(1,3,4,2)(5,8,10,19)(6,12,13,27)(7,16,18,30)(9,20,23,32)(11,24,25,35)(14,26,21,33)(15,29,28,36)(17,31,22,34);(6,13)(8,19)(11,25)(14,21)(15,28)(16,30)(20,32)(31,34)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17390);