Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#20768 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C24×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
778 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)(4,5)(6,9)(7,8)(10,11)(12,13,17,21,18,22,16,32)(14,23,27,34,25,37,26,19)(15,28,30,35,31,42,29,20)(24,38,41,43,39,33,40,36);(5,8)(9,11)(12,14,16,26,18,25,17,27)(13,19,21,23,22,34,32,37)(15,24,29,40,31,39,30,41)(20,33,28,36,35,38,42,43);(4,6,7,10)(5,9,8,11)(12,15,16,29,18,31,17,30)(13,20,21,28,22,35,32,42)(14,24,26,40,25,39,27,41)(19,33,23,36,34,38,37,43);(14,25)(19,34)(23,37)(24,39)(26,27)(33,38)(36,43)(40,41)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13756);