Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C3 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G | |
1328 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18);(3,5,7,12)(4,8,10,15)(6,13,14,18)(9,11,16,17);(5,12)(6,14)(8,15)(11,17)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26536);