Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29227 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
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)(20,21,23,25)(22,28,30,35)(24,32,29,27)(26,34,33,31);(4,6,8,7)(5,9,11,10)(12,14,15,18)(13,16,17,19)(21,25)(26,33)(27,32)(28,35);(4,6,8,7)(5,10,11,9)(14,18)(16,19)(20,22,29,34,23,30,24,31)(21,26,32,35,25,33,27,28);(4,7,8,6)(5,10,11,9)(21,25)(22,30)(27,32)(31,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22215);