Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#1271 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C22⋊C4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,9,10,7)(6,8)(12,15,16,17)(19,20,21,23,22,24,25,26);(3,5,6,10)(4,7,8,9)(11,12)(13,17)(14,16)(15,18)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 547);