Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#27682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13,15,17)(14,16,18,19)(20,21,24,26)(22,30,32,35)(23,29,25,28)(27,33,34,31);(4,6,8,7)(5,9,11,10)(12,14)(13,16)(15,18)(17,19)(21,26)(27,34)(28,29)(30,35);(14,18)(16,19)(20,22,24,32)(21,27,26,34)(23,31,25,33)(28,30,29,35);(4,7,8,6)(5,10,11,9)(20,23,24,25)(21,28,26,29)(22,31,32,33)(27,30,34,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20670);