Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,30)(8,19,22,12)(9,23,25,13)(11,24,27,17)(16,32,20,31)(21,29,36,34)(26,33,28,35)(37,38,39);(5,7,10,18)(6,11,14,27)(8,20,22,16)(9,24,25,17)(12,28,19,26)(13,30,23,15)(21,33,36,35)(29,31,34,32);(2,4)(5,8,9,21,10,22,25,36)(6,12,13,29,14,19,23,34)(7,16,17,33,18,20,24,35)(11,26,15,31,27,28,30,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5338);