Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#2249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C2×C3×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(2,4)(5,6)(7,15)(8,20)(9,12)(10,23)(11,17)(13,16)(14,21)(18,27)(19,34)(22,35)(24,33)(25,36)(26,31)(28,30)(29,32)(37,38,39);(5,7,16,30,10,19,22,33,9,18,31,36,21,32,8,17)(6,11,20,29,14,25,26,27,12,24,35,34,23,28,13,15);(1,2,3,4)(6,12)(11,24)(13,26)(14,23)(15,27)(20,35)(25,28)(29,34)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5540);