Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,19)(11,16)(12,24)(13,28)(14,17)(15,21)(18,26)(20,35)(22,36)(23,33)(25,30)(27,38)(29,34)(31,37)(32,39);(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,30)(17,31)(18,34)(19,24)(21,26)(23,37)(27,39)(28,32)(33,35)(36,38);(5,7)(8,11,14,13)(9,16,17,28)(10,20,22,29)(12,23,27,26)(15,30,31,32)(18,24,33,38)(19,35,36,34)(21,25,37,39);(9,17)(15,31)(16,28)(18,33)(19,36)(24,38)(30,32)(34,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26507);