Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
342 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 176 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:
(2,5)(3,7)(9,10,13,17)(11,14,19,16)(12,20,22,23)(15,18,24,21)(25,26,27);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,17,16)(12,21,22,18)(15,23,24,20);(2,5)(6,8)(9,12,13,22)(10,15,17,24)(11,18,19,21)(14,20,16,23);(1,3,4,7)(2,6,5,8)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17695);