Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#1624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(4,8)(5,6)(9,10)(12,15)(13,17)(16,18);(3,4,7,9)(5,8,6,10)(13,17)(15,18);(3,5,7,6)(4,8,9,10)(11,12,14,16)(13,18,17,15)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11496);