Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11033 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #394 |
| 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 | |
1264 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)(6,14,17,10)(7,18,20,11)(8,22)(9,12)(13,25)(15,26)(16,27,23,30)(19,35)(21,29,24,31)(28,34)(32,33);(4,6)(5,10)(7,19)(8,23)(9,17)(11,28)(12,14)(13,30)(15,21)(16,25)(18,35)(20,34)(22,27)(24,33)(26,29)(31,32);(4,7)(5,11)(6,15)(8,24)(9,20)(10,26)(12,18)(13,31)(14,32)(16,34)(17,33)(19,23)(21,25)(22,29)(27,35)(28,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4021);