Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C23×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 220 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)(17,32)(19,33)(20,30)(24,26)(25,29)(27,36)(28,35)(31,34)(37,38,39);(5,7,10,19)(6,11,15,28)(8,20,22,17)(9,24,25,36)(12,27,14,26)(13,30,31,32)(16,29,33,23)(18,34,35,21);(5,8,10,22)(6,12,15,14)(7,17,19,20)(9,21,25,34)(11,26,28,27)(13,29,31,23)(16,32,33,30)(18,24,35,36)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5432);