Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1315 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C8 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C22×C3 | C22⋊C8 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C8 ◁ G | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31)(41,42,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 451);