Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#2242 |
| 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 | |
138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,23)(12,17)(13,31)(14,34)(15,35)(16,22)(18,27)(19,24)(20,33)(21,32)(25,37)(26,36)(28,40)(29,39)(30,38)(41,42,43);(1,3,4,7)(2,5,6,8)(9,11,24,29,15,26,33,30,16,27,14,25,32,28,13,12)(10,17,31,40,21,37,34,18,22,38,20,36,35,39,19,23);(2,6)(5,8)(9,12,13,28,32,25,14,27,16,30,33,26,15,29,24,11)(10,18,19,37,35,40,20,17,22,23,34,39,21,36,31,38)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5533);