Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#2218 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
516 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)(6,9)(7,8)(10,11)(12,13)(14,19)(15,25)(16,30)(17,32)(18,22)(20,29)(21,27)(23,37)(24,36)(26,39)(28,40)(31,43)(33,38)(34,41)(35,42);(5,8)(9,11)(12,14,18,17)(13,19,22,32)(15,26,28,23)(16,24,31,27)(20,34,35,33)(21,30,36,43)(25,39,40,37)(29,41,42,38);(12,15,27,41,17,26,31,42,18,28,24,38,14,23,16,29)(13,20,30,37,19,33,36,40,22,35,43,39,32,34,21,25);(4,6,7,10)(5,9,8,11)(15,28)(20,35)(23,26)(25,40)(29,42)(33,34)(37,39)(38,41)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26738);