Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C2×C3×#3585 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G | |
11138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2)(19,20,21);(3,4)(5,9)(6,8)(7,10)(11,14)(12,16)(13,15)(17,18);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17);(3,5)(4,9)(6,13)(7,12)(8,15)(10,16)(11,17)(14,18);(5,12)(9,16)(13,18)(15,17);(4,10)(8,11)(9,16)(15,17)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56076);