Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×C4×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C42 ◁ G | |
360 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3,4)(13,14,15);(5,6,9,7,10,11,8,12);(5,7)(6,8)(9,11)(10,12)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5049);