Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23530 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#121 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,20)(11,25)(12,17)(13,18)(14,19)(15,23)(16,28)(21,30)(22,32)(24,31)(26,38)(27,34)(29,33)(35,36)(37,39);(8,10)(9,15)(11,26)(12,22)(13,23)(14,24)(16,30)(17,20)(18,31)(19,32)(21,29)(25,35)(27,37)(28,36)(33,39)(34,38);(5,7)(8,11,12,27,14,29,13,28)(9,16,18,33,19,34,17,25)(10,21,23,37,24,26,22,36)(15,30,20,35,32,38,31,39);(8,12,14,13)(9,17,19,18)(10,22,24,23)(11,27,29,28)(15,20,32,31)(16,25,34,33)(21,36,26,37)(30,35,38,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16518);