Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#1277 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D4 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle, prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ G | |
180 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 180 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) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 23 points:
(5,6,7,9,8,10,11,12)(14,18)(15,16)(19,20)(21,22,23);(1,2,3,4)(13,14,16,19,17,20,15,18)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 413);