Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#29654 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
948 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)(4,5,8,11)(6,12,14,9)(7,10,15,17)(13,18,19,16)(20,21,24,29)(22,30,32,26)(23,28,25,27)(31,35,33,34);(4,6,8,14)(5,9,11,12)(7,13,15,19)(10,16,17,18)(20,22,24,32)(21,26,29,30)(23,31,25,33)(27,35,28,34);(4,7)(5,10)(6,13)(8,15)(9,16)(11,17)(12,18)(14,19)(21,27)(23,25)(26,34)(28,29)(30,35)(31,33);(7,15)(10,17)(13,19)(16,18)(20,23,24,25)(21,28,29,27)(22,31,32,33)(26,34,30,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22642);