Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10753 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #385 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9,12,18,14,19,13,17)(10,20,24,29,23,30,22,15)(11,16,26,31,27,33,25,32)(21,35,36,28,37,34,38,39);(8,10,13,22,14,23,12,24)(9,15,17,30,19,29,18,20)(11,21,25,38,27,37,26,36)(16,28,32,35,33,39,31,34);(8,11)(9,16)(10,21)(12,26)(13,25)(14,27)(15,28)(17,32)(18,31)(19,33)(20,34)(22,38)(23,37)(24,36)(29,39)(30,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3741);