Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#27060 |
| 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 ◁ C3×C4×C22⋊C4 ◁ G | |
680 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)(6,10)(7,11)(8,9)(12,13)(14,18)(15,16)(17,19)(20,21,24,28)(22,29,32,26)(23,33,25,27)(30,35,31,34);(5,9)(10,11)(13,16)(17,18)(20,22,24,32)(21,26,28,29)(23,31,25,30)(27,34,33,35);(4,6,8,7)(5,10,9,11)(12,14,15,19)(13,17,16,18)(22,30)(23,25)(26,34)(27,33)(29,35)(31,32);(4,7,8,6)(5,10,9,11)(13,16)(17,18)(20,23,24,25)(21,27,28,33)(22,31,32,30)(26,34,29,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20048);