Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#2184 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,23)(15,20)(16,19)(17,30)(18,24)(21,29)(22,28)(25,31)(26,33)(27,35)(32,41)(34,39)(36,43)(37,42)(38,40);(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,24)(16,25)(17,26)(19,34)(20,23)(21,32)(22,33)(27,39)(28,40)(29,42)(30,38)(31,35)(36,37)(41,43);(6,11)(8,10)(13,19)(14,24)(16,29)(17,28)(18,31)(20,35)(21,22)(23,34)(25,41)(27,43)(30,36)(32,38)(33,37)(39,42);(5,9)(8,10)(13,20)(18,23)(19,35)(21,36)(22,30)(31,34)(32,37)(33,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26765);