Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(5,9)(10,11)(12,13)(14,17)(15,18)(16,19)(20,23)(21,24)(22,25)(26,27);(4,5,8,9)(6,10,7,11)(12,14,16,22)(13,17,19,25)(15,21,23,27)(18,24,20,26);(4,6,8,7)(5,10,9,11)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(4,7,8,6)(5,10,9,11);(13,19)(17,25)(18,20)(24,26)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54391);