Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C3×#3154 |
| 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 | |
3342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(4,6)(8,12)(10,17)(13,18)(14,19)(16,21)(20,22);(7,8,10,12,11,14,17,19)(9,13,16,18,15,20,21,22);(3,4,5,6)(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22);(9,15)(13,20)(16,21)(18,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53628);