Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#1841 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #383 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×M16 ◁ G | |
162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,19);(2,7,10,9)(8,15,16,11);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5132);