Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63065 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,9)(8,12)(10,11)(13,17)(15,19)(16,18)(20,21,26,27)(22,23,30,35)(24,28,25,29)(31,33,32,34);(5,11)(7,17)(12,14)(15,18)(21,27)(22,30)(28,29)(31,32);(4,6,10,8)(5,12,11,14)(7,15,17,18)(9,16,13,19)(20,22)(21,23)(24,31)(25,32)(26,30)(27,35)(28,33)(29,34);(4,7)(5,13)(6,15)(8,18)(9,11)(10,17)(12,19)(14,16)(20,23)(21,22)(24,33)(25,34)(26,35)(27,30)(28,31)(29,32);(4,8,10,6)(5,12,11,14)(7,18,17,15)(9,16,13,19)(20,24,26,25)(21,28,27,29)(22,31,30,32)(23,33,35,34);(4,9,10,13)(5,7,11,17)(6,16,8,19)(12,15,14,18)(20,25,26,24)(21,29,27,28)(22,32,30,31)(23,34,35,33)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56053);