Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(2,4)(5,6,10,14)(7,15,19,29)(8,12,20,26)(9,22,16,13)(11,23,25,18)(17,28,31,36)(21,33,30,27)(24,34,35,32)(37,38,39);(5,7,16,23,10,19,9,18)(6,11,22,29,14,25,13,15)(8,17,30,34,20,31,21,32)(12,24,33,36,26,35,27,28);(1,2,3,4)(5,8)(6,12)(7,17)(9,21)(10,20)(11,24)(13,27)(14,26)(15,28)(16,30)(18,32)(19,31)(22,33)(23,34)(25,35)(29,36);(8,20)(12,26)(17,31)(21,30)(24,35)(27,33)(28,36)(32,34)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17596);