Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11037 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #392 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | #1438 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,9,13,10,14,8,23)(6,15,17,11,18,27,24,26)(7,12,21,29,22,31,20,30)(16,28,33,35,34,32,19,25);(4,6)(5,11)(7,19)(8,24)(9,17)(10,18)(12,28)(13,27)(14,26)(15,23)(16,21)(20,34)(22,33)(25,30)(29,35)(31,32);(4,7)(5,12)(6,16)(8,20)(9,21)(10,22)(11,25)(13,29)(14,31)(15,32)(17,33)(18,34)(19,24)(23,30)(26,35)(27,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4025);