Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2132 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×Q8 | C24×C3 |
| socle | C22×C3 | C2×#99 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4,7,12,8,13,5,10)(6,11,14,18,15,9,16,17);(4,9)(6,15)(10,17)(11,13)(12,18)(14,16);(3,5,8,7)(4,10,13,12)(6,14,15,16)(9,17,11,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26353);