Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2)(3,4)(5,6,11,16)(7,17,21,28)(8,22,26,31)(9,15,10,14)(12,20,29,19)(13,25,32,24)(18,33,35,27)(23,30,34,36)(37,38,39);(2,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,33)(24,35)(26,34)(27,31)(32,36);(5,8,11,26)(6,13,16,32)(7,18,21,35)(9,25,10,24)(12,27,29,33)(14,31,15,22)(17,30,28,36)(19,23,20,34)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5311);