Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,13)(6,15,18,28)(7,19,22,12)(9,14,23,11)(10,20,26,16)(17,30,33,31)(21,27,35,29)(24,32,25,34);(4,6,8,18)(5,10,13,26)(7,20,22,16)(9,17,23,33)(11,24,14,25)(12,28,19,15)(21,34,35,32)(27,30,29,31);(7,21)(12,29)(15,30)(16,32)(19,27)(20,34)(22,35)(28,31);(5,11)(10,24)(12,27)(13,14)(15,31)(19,29)(25,26)(28,30)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11307);