Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #327 |
| commutator | C2×C8 | C2×C3×C4 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,10,7,11,6,22)(3,12,14,26,16,29,15,32)(4,17,19,9,21,27,20,28)(8,18,23,31,25,30,24,13)(33,34,35);(1,3,7,16)(2,8,11,25)(4,18,21,30)(5,15,6,14)(9,26,28,32)(10,23,22,24)(12,27,29,17)(13,20,31,19)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 478);