Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×#62988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
11186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 23 points:
(1,2,3)(5,7)(8,9,13,14)(10,15,20,23)(11,16,21,19)(12,17,22,18);(4,5,6,7);(9,14)(12,22)(15,23)(16,19);(8,10)(9,15)(11,17)(12,19)(13,20)(14,23)(16,22)(18,21);(8,11)(9,16)(10,18)(12,15)(13,21)(14,19)(17,20)(22,23);(8,12)(9,17)(10,19)(11,15)(13,22)(14,18)(16,20)(21,23)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55976);