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×#33776 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
592 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)(6,10)(7,9)(8,11)(12,13,18,22)(14,23,26,41)(15,20,17,32)(16,28,31,42)(19,34,37,25)(21,27,39,30)(24,40,33,38)(29,35,43,36);(4,6,7,11)(5,8,9,10);(6,11)(8,10)(12,14)(13,19)(15,24)(16,29)(17,33)(18,26)(20,35)(21,38)(22,37)(23,28)(25,30)(27,34)(31,43)(32,36)(39,40)(41,42);(5,9)(8,10)(12,15,18,17)(13,20,22,32)(14,24,26,33)(16,27,31,30)(19,35,37,36)(21,28,39,42)(23,40,41,38)(25,29,34,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26764);