Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C2×C3×#1274 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C42 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C23×C3 | #95 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(27,28,29);(3,4,5,7,6,8,9,10)(11,12)(13,19,21,25)(14,17)(15,18)(16,20,23,26)(22,24);(11,13)(12,16,18,23)(14,20)(15,21)(17,19,24,25)(22,26)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 841);