Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2138 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C2×D8 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
228 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 104 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)(7,16)(8,12)(9,23)(10,15)(11,18)(13,21)(14,22)(17,29)(19,35)(20,32)(24,31)(25,28)(26,33)(27,34)(30,36)(37,38,39);(5,7)(6,11)(8,17)(9,24)(10,20)(12,26)(13,29)(14,31)(15,27)(16,23)(18,21)(19,22)(25,33)(28,32)(30,35)(34,36);(5,8,10,22)(6,12,15,14)(7,17,20,19)(9,21,25,36)(11,26,27,31)(13,28,30,23)(16,29,32,35)(18,33,34,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5429);