Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
808 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)(7,9)(8,11)(12,13)(14,18)(15,16)(17,19)(20,21,25,33,24,28,23,27)(22,29,31,26,30,35,32,34);(4,6,7,11)(5,8,9,10)(13,16)(14,19)(22,30)(26,34)(29,35)(31,32);(12,14,15,19)(13,17,16,18)(20,22,24,30)(21,26,28,34)(23,31,25,32)(27,35,33,29);(5,9)(8,10)(13,16)(17,18)(20,23,24,25)(21,27,28,33)(22,31,30,32)(26,35,34,29)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22321);