Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(3,4,5);(6,7,11,16)(8,17,21,33)(9,22,18,14)(10,13,24,15)(12,25,28,19)(20,31,36,32)(23,30,34,29)(26,37,27,35);(1,2)(6,8,18,25,11,21,9,19)(7,12,22,33,16,28,14,17)(10,20,34,37,24,36,23,35)(13,26,30,32,15,27,29,31);(1,2)(7,13)(10,24)(12,26)(14,29)(15,16)(17,31)(20,36)(22,30)(23,34)(27,28)(32,33)(35,37)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17506);