Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×#60113 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C23×D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C2×C3 | C23×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×#41 ◁ G | |
2564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,9,13,10,14,8,23)(6,15,19,25,17,29,18,11)(7,12,22,27,20,26,21,33)(16,28,32,34,30,35,31,24);(4,6,8,18,10,17,9,19)(5,11,13,15,14,25,23,29)(7,16,21,31,20,30,22,32)(12,24,27,28,26,34,33,35);(4,7)(5,12)(6,16)(8,21)(9,22)(10,20)(11,24)(13,27)(14,26)(15,28)(17,30)(18,31)(19,32)(23,33)(25,34)(29,35);(7,20)(12,26)(16,30)(21,22)(24,34)(27,33)(28,35)(31,32);(6,17)(11,25)(15,29)(16,30)(18,19)(24,34)(28,35)(31,32)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53101);