Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C3×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×Q8 ◁ C2×C3×2−1+4 ◁ G | |
1188 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)(6,11)(7,12)(8,13,21,10)(9,14)(15,24)(16,25,31,22)(17,26)(18,27)(19,28,33,23)(20,29)(30,35,32,34);(5,10)(8,21)(11,22)(12,23)(13,14)(16,31)(19,33)(24,34)(25,26)(27,35)(28,29)(30,32);(4,6,9,17)(5,11,14,26)(7,18,20,15)(8,16,21,31)(10,22,13,25)(12,27,29,24)(19,32,33,30)(23,35,28,34);(4,7,9,20)(5,12,14,29)(6,15,17,18)(8,19,21,33)(10,23,13,28)(11,24,26,27)(16,30,31,32)(22,34,25,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26348);