Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#11070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C25×C3 ◁ G | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,7)(8,9,10,11)(12,13)(14,17)(15,18)(16,19)(20,25)(21,24)(22,26)(23,27);(4,5,6,7)(12,14)(13,17)(15,20)(16,21)(18,25)(19,24)(22,27)(23,26);(12,15,16,22)(13,18,19,26)(14,20,21,27)(17,23,24,25)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4058);