Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#2911 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
492 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 344 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 — 4 generators on 23 points:
(1,2,3,4)(5,6)(7,14)(8,11)(9,13)(10,16)(12,17)(15,19)(18,20)(21,22,23);(5,7,8,16,9,15,17,20)(6,10,12,14,13,18,11,19);(7,15)(10,18)(14,19)(16,20);(5,8,9,17)(6,11,13,12)(7,16,15,20)(10,19,18,14)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17351);