Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
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)(20,21,25,30)(22,26,31,35)(23,28,33,34)(24,27,32,29);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22,25,31)(21,26,30,35)(23,32,33,24)(27,28,29,34);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,27)(22,31)(24,32)(26,34)(28,35)(29,30);(6,14)(9,12)(13,19)(16,18)(20,23)(21,28)(22,32)(24,31)(25,33)(26,29)(27,35)(30,34);(5,11)(9,12)(10,17)(16,18)(22,31)(23,33)(26,35)(28,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54784);