Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3202 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ G | |
378 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 168 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 — 4 generators on 43 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,29)(12,22,24,32)(13,25,27,17)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35)(41,42,43);(1,3)(2,5)(4,8)(6,7)(9,11,14,20)(10,15,18,29)(12,19,24,34)(13,21,27,36)(16,28,31,39)(17,30,25,37)(22,33,32,40)(23,35,26,38);(3,8)(5,7)(9,12,14,24)(10,16,18,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(17,32,25,22)(21,38,36,35)(30,40,37,33);(2,6)(5,7)(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17642);