Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#1670 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(4,7)(5,9)(8,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21);(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(3,4,6,8)(5,10,9,7)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12101);