Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#1979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
276 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,4,3)(5,6,10,15,11,14,9,23)(7,12,18,27,17,28,24,26)(8,19,21,29,22,33,20,13)(16,30,32,25,31,35,34,36)(37,38,39);(1,3,4,2)(5,7,9,18,11,17,10,24)(6,12,15,26,14,28,23,27)(8,16,20,32,22,31,21,34)(13,25,19,30,29,36,33,35);(5,8,10,21,11,22,9,20)(6,13,23,33,14,29,15,19)(7,16,24,34,17,31,18,32)(12,25,27,35,28,36,26,30)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5270);