Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#2901 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C2×C3 | C22×C22⋊C4 |
| prime radical | C3×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×#41 ◁ G | |
738 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 19 points:
(1,2,6,3,7,9,5,11)(4,8,14,10,12,15,13,16)(17,18,19);(1,3)(2,5)(4,10)(6,9)(7,11)(8,13)(12,16)(14,15);(1,4)(2,8)(3,10)(5,13)(6,14)(7,12)(9,15)(11,16);(4,12)(8,15)(10,16)(13,14)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17341);