Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61829 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C23×C3 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G | |
1650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,24,27)(22,28,29,35)(23,31,33,26)(25,32,34,30);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(22,29)(25,34)(28,35)(30,32);(20,22,24,29)(21,25,27,34)(23,32,33,30)(26,35,31,28);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18);(5,11)(9,12)(10,19)(17,18)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54817);