Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3508 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13)(6,14,17,18)(7,11,8,12)(10,16,19,15)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30);(4,6)(5,10)(7,15)(8,16)(9,17)(11,18)(12,14)(13,19)(23,32)(26,31)(28,35)(30,33);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,18,19,14)(20,22)(21,25)(23,33)(24,29)(26,35)(27,34)(28,31)(30,32);(4,7,9,8)(5,12,13,11)(6,16,17,15)(10,18,19,14)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11623);