Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10771 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #381 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | C22⋊#95 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,10,15,23)(7,17,18,11)(8,20)(9,12)(13,21)(14,26)(16,30,29,24)(19,25,32,31)(22,27)(28,33)(34,35);(4,6,8,16,9,15,21,29)(5,10,13,24,12,23,20,30)(7,14,19,28,18,27,32,35)(11,22,25,33,17,26,31,34);(4,7)(5,11)(6,14)(8,19)(9,18)(10,22)(12,17)(13,25)(15,27)(16,28)(20,31)(21,32)(23,26)(24,33)(29,35)(30,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3759);