Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(20,21,23,27,24,28,31,26)(22,25,30,34,29,33,35,32);(5,8)(9,10)(12,14,15,18)(13,16,17,19)(20,22)(21,25)(23,30)(24,29)(26,32)(27,34)(28,33)(31,35);(14,18)(16,19)(21,26)(23,31)(25,32)(27,28)(30,35)(33,34);(4,6,7,11)(5,9,8,10)(22,29)(25,33)(30,35)(32,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22121);