Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#1426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #367 |
| commutator | C16 | C2×C3×C4 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| prime radical | C22×C3 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(2,7,9,20)(3,12)(4,16)(5,17)(8,23)(10,24,27,32)(11,30,28,22)(13,18)(14,26)(15,25)(19,31,29,21)(33,34,35);(1,2,8,24,4,10,3,7,17,28,12,21,13,19,23,30,6,9,26,32,18,27,15,20,5,11,25,31,16,29,14,22)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 562);