Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17576 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C3×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,13,16,10)(7,17)(8,11)(9,12)(14,18,15,19)(20,21)(22,30,32,35)(23,28)(24,29)(25,27)(26,31,34,33);(4,6)(5,10)(7,14)(8,15)(9,16)(11,18)(12,13)(17,19)(20,22)(21,26)(23,31)(24,33)(25,32)(27,34)(28,30)(29,35);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,18,13,19)(21,27)(26,34)(28,29)(30,35);(4,8,9,7)(5,11,12,17)(6,15,16,14)(10,18,13,19)(20,23,25,24)(21,28,27,29)(22,31,32,33)(26,30,34,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10564);