Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C1 |
| direct product | C3×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | D4 |
| commutator | C4 | C2×C3×C16 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| socle | C2×C3 | C4⋊C16 |
| prime radical | C22×C3 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C2×C3×C16 ◁ G | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,10,6,11,20,27,4,12,18,28,19,24,31,32,7,13,21,29,3,14,16,30,17,9,25,26,15,8,22,23)(33,34,35);(1,3,4,15,7,6,17,19)(2,8,9,14,13,24,12,11)(5,16,18,22,21,20,25,31)(10,23,26,30,29,32,28,27)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 558);