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×#3122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
2338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
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)(13,20)(14,18)(15,17)(16,21)(19,23)(22,25)(24,26);(4,7)(8,10)(11,13)(12,16)(14,21)(15,22)(17,24)(18,20)(19,25)(23,26);(12,17)(15,23)(16,24)(18,19)(20,25)(22,26);(3,5,6,9)(4,8,7,10)(13,21)(16,20)(22,26)(24,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53539);