Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33516 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,19)(11,16)(12,24)(13,28)(14,18)(15,21)(17,26)(20,35)(22,36)(23,34)(25,30)(27,38)(29,33)(31,39)(32,37);(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,30)(17,33)(18,32)(19,24)(21,26)(23,37)(27,39)(28,31)(34,35)(36,38);(5,7);(8,11,14,13)(9,16,18,28)(10,20,22,29)(12,23,27,26)(15,30,32,31)(17,24,34,38)(19,35,36,33)(21,25,37,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26504);