Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#2835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
380 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 168 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 — 4 generators on 27 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,14)(13,16)(19,23)(21,24)(25,26,27);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24);(3,7)(5,8)(9,11,13,12)(10,15,16,14)(17,19,20,23)(18,21,22,24);(9,12)(10,14)(11,13)(15,16)(18,22)(21,24)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17275);