Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3099 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G | |
2234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(11,12,16,20)(13,17,22,26)(14,19,15,24)(18,23,25,21);(4,7)(8,10)(11,13)(12,17)(14,23)(15,21)(16,22)(18,24)(19,25)(20,26);(3,5,6,9)(4,8,7,10)(11,13)(12,18)(14,21)(15,23)(16,22)(17,24)(19,26)(20,25);(11,14,16,15)(12,19,20,24)(13,21,22,23)(17,18,26,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53503);