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) = C22 |
| direct product | C3×#1376 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | D4 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C22×C3 | C4⋊C8 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C8 ◁ G | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(5,6,8,12,7,10,14,17,9,13,16,20,15,11,18,19)(21,22,23);(1,2,3,4)(6,10,13,11)(12,17,20,19)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 512);