Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C3×#1517 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,9)(6,8)(7,10)(11,12)(13,19,21,16)(14,17)(15,18)(20,25,22,24)(23,26);(4,8)(5,6)(9,10)(11,13)(12,16)(14,22)(15,21)(17,25)(18,19)(20,23)(24,26);(3,5,7,6)(4,8,10,9)(11,14,15,23)(12,17,18,26)(13,20,21,22)(16,24,19,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10557);