Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
508 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)(14,22)(15,26)(16,27)(18,29)(20,35)(21,34)(23,28)(24,39)(25,38)(30,33)(31,40)(32,43)(36,37)(41,42);(5,8)(9,11)(12,13,17,19)(14,18,25,31)(15,20,28,33)(16,21,27,34)(22,29,38,40)(23,30,26,35)(24,32,36,41)(37,42,39,43);(12,14,23,39,16,24,15,22,17,25,26,37,27,36,28,38)(13,18,30,43,21,32,20,29,19,31,35,42,34,41,33,40);(4,6,7,10)(5,9,8,11)(13,19)(18,31)(20,33)(21,34)(29,40)(30,35)(32,41)(42,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26731);