Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#2103 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #400 |
| prime radical | C23×C3 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,10,6,11,7,22)(3,12,14,24,16,27,15,8)(4,17,19,31,20,9,21,26)(13,25,28,32,29,30,18,23)(33,34,35);(1,3,7,14,6,16,5,15)(2,8,22,12,11,24,10,27)(4,18,21,13,20,28,19,29)(9,25,31,32,17,30,26,23);(1,4,5,19,6,20,7,21)(2,9,10,26,11,17,22,31)(3,13,14,28,16,29,15,18)(8,23,12,25,24,32,27,30)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5394);