Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61812 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,26,31)(22,27,32,28)(23,29,33,35)(24,30,25,34);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,27)(23,25)(24,33)(26,32)(28,31)(29,34)(30,35);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(20,23)(21,28)(22,25)(24,32)(26,33)(27,31)(29,34)(30,35);(6,14)(9,12)(13,19)(16,18)(20,23)(21,29)(22,24)(25,32)(26,33)(27,30)(28,34)(31,35);(5,11)(9,12)(10,17)(16,18)(20,24,26,25)(21,30,31,34)(22,23,32,33)(27,29,28,35)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54800);