Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,10,15)(7,16,20,34)(8,12,22,14)(9,23,25,29)(11,26,28,18)(13,30,32,21)(17,31,19,36)(24,33,35,27)(37,38,39);(5,7)(6,11)(8,17)(9,24)(10,20)(12,27)(13,31)(14,33)(15,28)(16,23)(18,21)(19,22)(25,35)(26,30)(29,34)(32,36);(2,4)(5,8,10,22)(6,12,15,14)(7,17,20,19)(9,21,25,30)(11,27,28,33)(13,29,32,23)(16,31,34,36)(18,35,26,24)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5480);