Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C3×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(39,40,41);(4,6)(7,8,12,16)(9,17,21,31)(10,14,22,28)(11,24,18,15)(13,25,27,20)(19,30,33,38)(23,35,32,29)(26,36,37,34);(7,9,18,25,12,21,11,20)(8,13,24,31,16,27,15,17)(10,19,32,36,22,33,23,34)(14,26,35,38,28,37,29,30);(3,4,5,6)(7,10)(8,14)(9,19)(11,23)(12,22)(13,26)(15,29)(16,28)(17,30)(18,32)(20,34)(21,33)(24,35)(25,36)(27,37)(31,38);(10,22)(14,28)(19,33)(23,32)(26,37)(29,35)(30,38)(34,36)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53630);