Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C22×C3×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,4)(5,6)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(21,22,23);(6,10)(7,15)(8,17)(11,19)(12,13)(14,18);(2,4)(5,7,8,16,9,15,17,20)(6,11,12,19,13,14,10,18)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5336);