Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33762 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
588 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)(14,19)(15,25)(16,29)(17,33)(18,22)(20,27)(21,31)(23,39)(24,38)(26,37)(28,42)(30,34)(32,41)(35,43)(36,40);(4,6)(5,8)(7,11)(9,10)(12,14,18,17)(13,19,22,33)(15,26,28,23)(16,24,32,31)(20,35,36,34)(21,29,38,41)(25,37,42,39)(27,43,40,30);(6,11)(8,10)(12,15,18,28)(13,20,22,36)(14,23,17,26)(16,30,32,43)(19,34,33,35)(21,37,38,39)(24,40,31,27)(25,41,42,29);(5,9)(8,10)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26750);