Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10252 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,15)(8,11)(9,13)(10,17)(14,19)(16,18)(20,21,24,27)(22,28,30,35)(23,31,33,26)(25,32,34,29);(4,6,8,14)(5,9,11,18)(7,16,17,13)(10,19,15,12);(7,17)(9,18)(10,15)(13,16)(20,22,24,30)(21,25,27,34)(23,32,33,29)(26,35,31,28)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3240);