Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9245 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
456 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)(4,5,7,11,9,13,8,12)(6,14,15,10,16,18,17,19)(21,25,27,30)(22,28,29,31);(4,6)(5,10,13,19)(7,15)(8,17)(9,16)(11,18,12,14)(20,21,23,30,24,27,32,25)(22,26,31,34,29,35,28,33);(5,11)(7,8)(10,18)(12,13)(14,19)(15,17)(20,22)(21,26)(23,31)(24,29)(25,33)(27,35)(28,32)(30,34)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2233);