Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C24×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(39,40,41);(3,4,5,6)(7,8)(9,18)(10,15)(11,24)(12,28)(13,17)(14,20)(16,26)(19,32)(21,38)(22,35)(23,33)(25,31)(27,34)(29,36)(30,37);(7,9,13,22)(8,14,17,30)(10,21,12,19)(11,25,27,36)(15,31,28,29)(16,32,33,38)(18,34,35,24)(20,23,37,26);(7,10,13,12)(8,15,17,28)(9,19,22,21)(11,23,27,26)(14,29,30,31)(16,24,33,34)(18,32,35,38)(20,36,37,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26366);