Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9066 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #120 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9,13,15,12,10,11,14)(16,17,20,22,21,25,19,24)(18,26,28,23,29,30,27,31);(4,5,6,7)(8,10,12,9)(11,14,13,15)(17,22)(19,20)(23,26)(24,25)(27,28)(30,31);(11,13)(14,15)(16,18,21,29)(17,23,25,31)(19,28,20,27)(22,26,24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2054);