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×#33568 |
| 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 | |
1110 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)(6,15)(7,19)(8,11)(9,14)(10,17)(12,22)(13,24)(16,30)(18,32)(20,27)(21,29)(23,33)(25,31)(26,34)(28,35);(4,6)(5,10)(7,20)(8,23)(9,16)(11,26)(12,30)(13,28)(14,27)(15,19)(17,22)(18,24)(21,34)(25,35)(29,32)(31,33);(5,11)(8,24)(10,26)(12,31)(13,14)(15,32)(18,23)(19,29)(21,25)(27,28)(30,33)(34,35);(6,16)(10,27)(15,30)(17,20)(18,23)(26,28)(32,33)(34,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26556);