Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11278 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #387 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1442 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
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)(10,20,22,35,23,15,29,31)(11,24,27,34,26,16,28,33)(21,30,38,39,37,36,25,32);(8,10)(9,15)(11,25,26,38)(12,22)(13,29)(14,23)(16,32,24,39)(17,20)(18,31)(19,35)(21,27,37,28)(30,33,36,34);(8,11)(9,16,17,24)(10,21)(12,28)(13,27)(14,26)(15,30,20,36)(18,34,19,33)(22,38)(23,37)(25,29)(31,39,35,32)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4266);