Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#20884 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,13,18,14,17,12,26)(10,19,21,15,22,31,27,30)(11,23,24,16,25,33,28,32)(20,29,36,39,37,38,35,34);(5,7)(8,10,13,21,14,22,12,27)(9,15,18,31,17,30,26,19)(11,20,24,36,25,37,28,35)(16,29,33,39,32,38,23,34);(8,11)(9,16)(10,20)(12,28)(13,24)(14,25)(15,29)(17,32)(18,33)(19,34)(21,36)(22,37)(23,26)(27,35)(30,38)(31,39);(9,17)(15,30)(16,32)(18,26)(19,31)(23,33)(29,38)(34,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13872);