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×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,2,3)(4,5,6,7)(8,11,10,9);(4,8,6,10)(5,9,7,11);(12,13)(14,18)(15,16)(17,19);(13,16)(17,18);(12,13)(14,17)(15,16)(18,19);(12,14)(13,17)(15,19)(16,18)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55970);