Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C2×C3×#3021 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
4934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(19,20,21);(4,9)(6,7)(10,17)(11,14)(12,15)(16,18);(3,4)(5,10)(6,9)(7,14)(8,11)(12,16)(13,18)(15,17);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,18)(14,17);(3,6,8,7)(4,9,11,14)(5,12,13,15)(10,16,18,17)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53373);