Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,13,19,14,17,12,18)(10,20,23,30,24,15,22,31)(11,16,26,34,27,33,28,35)(21,36,25,39,37,29,38,32);(8,10,14,24)(9,15,17,20)(11,25,27,38)(12,23,13,22)(16,32,33,39)(18,31,19,30)(21,28,37,26)(29,35,36,34);(5,7)(8,11)(9,16)(10,21)(12,28)(13,26)(14,27)(15,29)(17,33)(18,35)(19,34)(20,36)(22,38)(23,25)(24,37)(30,39)(31,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4003);