Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#17841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C4⋊C4 |
| prime radical | C2×C7×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×#41 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,15)(12,22)(13,16)(14,20)(18,19)(21,23)(24,25,28,32,29,33,27,31)(26,34,37,38,35,39,36,30);(8,10,11,19,13,21,12,20)(9,14,22,23,16,18,15,17)(24,26,28,37,29,35,27,36)(25,30,31,39,33,38,32,34);(8,9)(10,18)(11,22)(12,15)(13,16)(14,21)(17,19)(20,23)(26,35)(30,38)(34,39)(36,37);(8,11,13,12)(9,15,16,22)(10,19,21,20)(14,17,18,23)(24,27,29,28)(25,31,33,32)(26,36,35,37)(30,39,38,34)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10829);