Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C8 |
| direct product | C3×#1720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 186 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20)(25,26,27);(1,3,4,6,5,7,8,2)(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17);(1,2,4,3,5,6,8,7)(11,19)(14,23)(17,24)(20,22)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5011);