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) = C213 |
| direct product | C3×Q8×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| prime radical | C2×C3×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5,6,7)(8,11,10,9);(4,8,6,10)(5,9,7,11);(13,17)(15,23)(18,20)(21,27);(12,13,16,17)(14,20,22,18)(15,19,23,24)(21,26,27,25);(12,14,16,22)(13,18,17,20)(15,21,23,27)(19,25,24,26);(12,15)(13,19)(14,21)(16,23)(17,24)(18,25)(20,26)(22,27)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55985);