Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C32 · Schur multiplier (H2) = C2 |
| direct product | C3×#7510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C32 | C22 |
| commutator | C2 | C3×C4×C32 |
| Frattini, cosocle | C2×C32 | C22×C3 |
| socle, prime radical | C22×C3 | C2×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C32 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3)(8,9,10,16,11,17,20,34,12,18,21,35,24,37,40,56,13,19,22,36,25,38,41,57,27,39,43,58,46,59,60,69,14,15,23,30,26,31,42,50,28,32,44,51,47,53,61,65,29,33,45,52,48,54,62,66,49,55,63,67,64,68,70,71);(4,5,6,7)(9,15)(16,30)(17,31)(18,32)(19,33)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(56,65)(57,66)(58,67)(59,68)(69,71)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 498);