Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#1723 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
176 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,5)(3,7)(6,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19)(25,26,27);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5014);