Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1430 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)(6,14)(7,19)(8,22)(9,13)(10,17)(11,20)(12,23)(15,28)(16,29)(18,27)(21,32)(24,31)(25,33)(26,30)(34,35);(4,6,8,17,9,18,23,28)(5,10,12,14,13,15,22,27)(7,16,20,30,21,31,33,35)(11,24,19,26,25,29,32,34);(4,5)(6,15,18,10)(7,11)(8,12)(9,13)(14,17,27,28)(16,29,31,24)(19,20)(21,25)(22,23)(26,30,34,35)(32,33);(4,7,8,20,9,21,23,33)(5,11,12,19,13,25,22,32)(6,16,17,30,18,31,28,35)(10,24,14,26,15,29,27,34)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18786);