Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13,18,23)(14,19,26,37)(15,21,17,33)(16,29,32,43)(20,38,40,25)(22,28,41,31)(24,34,27,36)(30,35,42,39);(4,6,7,11)(5,8,9,10)(12,14)(13,19)(15,27)(16,25)(17,24)(18,26)(20,29)(21,36)(22,35)(23,37)(28,42)(30,31)(32,38)(33,34)(39,41)(40,43);(12,14)(13,20)(15,24)(16,30)(17,27)(18,26)(19,29)(21,35)(22,36)(23,40)(25,31)(28,38)(32,42)(33,39)(34,41)(37,43);(12,15,18,17)(13,21,23,33)(14,24,26,27)(16,28,32,31)(19,34,37,36)(20,35,40,39)(22,29,41,43)(25,30,38,42)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26622);