Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#2251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C2×C3×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(2,4)(5,6,10,14)(7,15,17,29)(8,20,22,35)(9,23,21,13)(11,24,25,18)(12,27,28,16)(19,34,33,31)(26,32,30,36)(37,38,39);(5,7,16,32,9,19,22,24,10,17,27,36,21,33,8,18)(6,11,20,31,13,26,28,29,14,25,35,34,23,30,12,15);(1,2,3,4)(7,17)(11,25)(15,29)(18,24)(19,33)(26,30)(31,34)(32,36)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5542);