Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #394 |
| 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 | |
1238 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)(8,9,13,18,14,17,12,29)(10,19,22,34,23,15,21,31)(11,24,28,35,26,16,27,33)(20,30,38,32,37,36,25,39);(8,10,14,23)(9,15,17,19)(11,25,26,38)(12,22,13,21)(16,32,24,39)(18,34,29,31)(20,28,37,27)(30,35,36,33);(5,7)(8,11)(9,16)(10,20)(12,28)(13,27)(14,26)(15,30)(17,24)(18,35)(19,36)(21,38)(22,25)(23,37)(29,33)(31,39)(32,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4023);