Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1364 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #102 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(27,28,29);(4,7)(5,9)(8,10)(11,12,13,17,21,25,26,24,15,19,22,23,14,18,20,16);(3,4,6,8)(5,10,9,7)(12,16)(13,20)(14,21)(17,18)(19,24)(22,26)(23,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4925);