Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,28)(12,22,25,37)(13,26,23,17)(16,30,31,24)(19,32,34,40)(21,36,33,29)(27,38,39,35)(41,42,43);(2,5)(6,8)(9,11,14,20)(10,15,18,28)(12,19,25,34)(13,21,23,33)(16,27,31,39)(17,29,26,36)(22,32,37,40)(24,35,30,38);(9,12,23,30,14,25,13,24)(10,16,26,37,18,31,17,22)(11,19,33,38,20,34,21,35)(15,27,36,40,28,39,29,32);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(19,34)(21,33)(27,39)(29,36)(32,40)(35,38)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17617);