Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(5,8)(6,10)(12,13,17,21,16,20,30,37)(14,22,25,39,24,18,32,34)(15,26,29,38,28,19,31,36)(23,33,40,42,27,35,41,43);(4,5)(6,9)(7,8)(10,11)(12,14)(13,18)(15,27)(16,24)(17,25)(19,35)(20,22)(21,34)(23,28)(26,33)(29,41)(30,32)(31,40)(36,43)(37,39)(38,42);(5,8)(9,11)(12,15,16,28)(13,19,20,26)(14,23,24,27)(17,31,30,29)(18,33,22,35)(21,38,37,36)(25,41,32,40)(34,43,39,42);(4,6,7,10)(5,9,8,11)(13,20)(14,24)(19,26)(21,37)(23,27)(25,32)(36,38)(40,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14871);