Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#3549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,19);(3,8)(5,7)(9,10,13,14)(11,16,12,15);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,14,16);(2,6)(5,7)(9,11,13,12)(10,15,14,16);(9,12,13,11)(10,15,14,16)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20009);