Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#1366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #333 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| prime radical | C22×C3 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,17,10,28,26,6,11,19,27,5,8,18,24)(3,12,14,25,20,23,16,31,13,29,30,32,15,7,21,22)(33,34,35);(1,3,6,13)(2,7)(4,16,19,21)(5,20,17,15)(8,12)(9,25)(10,29)(11,23)(14,18,30,28)(22,26)(24,31)(27,32)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 502);