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×#33569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C3 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G | |
1082 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)(8,21)(9,24)(10,15)(11,17)(12,20)(14,26)(18,31)(19,34)(22,28)(23,30)(25,27)(29,35)(32,33);(4,6)(5,11)(7,17)(8,22)(9,25)(10,18)(12,27)(13,31)(14,29)(15,28)(16,21)(19,26)(20,35)(23,33)(24,32)(30,34);(4,7,10,8)(5,12,15,30)(6,17,18,22)(9,23,26,20)(11,27,28,34)(13,14,21,24)(16,32,31,29)(19,35,25,33);(6,18)(11,28)(16,31)(17,22)(19,25)(27,34)(29,32)(33,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26557);