Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C3 |
| prime radical | C3×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
1084 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)(6,16)(7,13)(8,22)(9,12)(10,15)(11,18)(14,25)(17,28)(19,32)(20,31)(21,29)(23,30)(24,33)(26,35)(27,34);(4,6,10,20)(5,11,15,28)(7,18,8,17)(9,24,25,19)(12,26,14,27)(13,31,22,16)(21,34,23,35)(29,33,30,32);(5,12)(9,25)(11,26)(13,29)(14,15)(16,32)(19,24)(21,23)(22,30)(27,28)(31,33)(34,35);(4,7,10,8)(5,13,15,22)(6,17,20,18)(9,21,25,23)(11,16,28,31)(12,29,14,30)(19,34,24,35)(26,32,27,33)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26559);