Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2)(43,44,45);(3,4)(5,8)(6,7)(9,10)(11,12,16,20)(13,17,22,30)(14,24,27,39)(15,28,25,19)(18,32,33,26)(21,34,36,42)(23,38,35,31)(29,40,41,37);(4,7)(8,10)(11,13,16,22)(12,17,20,30)(14,21,27,36)(15,23,25,35)(18,29,33,41)(19,31,28,38)(24,34,39,42)(26,37,32,40);(11,14,25,32,16,27,15,26)(12,18,28,39,20,33,19,24)(13,21,35,40,22,36,23,37)(17,29,38,42,30,41,31,34);(3,5,6,9)(4,8,7,10)(13,22)(17,30)(21,36)(23,35)(29,41)(31,38)(34,42)(37,40)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53669);