Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×C4×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C4⋊C4 |
| prime radical | C23×C3 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C42 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(21,22,23);(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5068);