Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#358 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
232 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(3,4,5);(6,7,8,9)(11,15)(12,18)(13,20)(16,25)(17,24)(19,22)(21,23);(6,7)(8,9)(10,11,14,17)(12,19,21,25)(13,22,20,16)(15,23,24,18)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5428);