Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×D4 ◁ C2×C3×H(5,𝔽2) ◁ G | |
1252 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)(6,14,17,27)(7,11)(8,20,22,12)(9,13)(10,16)(15,28,29,35)(18,25)(19,32,31,26)(21,24)(23,30)(33,34);(4,6,9,17)(5,10,13,24)(7,15,18,29)(8,21,22,16)(11,23,25,34)(12,27,20,14)(19,33,31,30)(26,35,32,28);(4,7)(5,11)(6,15)(8,19)(9,18)(10,23)(12,26)(13,25)(14,28)(16,30)(17,29)(20,32)(21,33)(22,31)(24,34)(27,35);(7,18)(11,25)(15,29)(19,31)(23,34)(26,32)(28,35)(30,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26349);