Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#17386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#538 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4×D4 ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,7,12,9,13,18,26)(6,11,16,24,14,25,28,35)(8,10,17,21,19,22,31,33)(15,20,29,32,27,23,30,34);(5,10)(6,14)(7,17)(11,23)(13,22)(15,27)(16,30)(18,31)(20,25)(24,35)(28,29)(32,34);(4,6)(5,11)(7,16)(8,15)(9,14)(10,20)(12,24)(13,25)(17,29)(18,28)(19,27)(21,32)(22,23)(26,35)(30,31)(33,34);(6,15)(11,20)(14,27)(16,29)(23,25)(24,32)(28,30)(34,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10374);