Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#2372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
560 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 280 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,17)(18,20)(21,22,23);(1,3,4,2)(13,14,16,17)(15,18,19,20);(6,9)(10,12)(13,15,16,19)(14,18,17,20);(5,7,8,11)(6,10,9,12)(14,17)(18,20)150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16812);