Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#1280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22⋊C4 |
| commutator | C22 | C3×C4×C8 |
| Frattini, cosocle | C23×C4 | C22×C3 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 2 generators on 23 points:
(5,6,7,9,8,10,11,12)(13,14)(15,19,20,17)(16,18)(21,22,23);(1,2,3,4)(13,15)(14,17)(16,20)(18,19)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 416);