Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #459 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1802 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
400 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)(8,9,14,18,15,20,12,28)(10,19,13,29,23,36,22,16)(11,24,26,39,27,31,25,35)(17,30,32,38,34,37,33,21);(8,10,12,22,15,23,14,13)(9,16,18,19,20,29,28,36)(11,21,25,37,27,38,26,30)(17,31,32,35,34,24,33,39);(8,11)(9,17)(10,21)(12,25)(13,30)(14,26)(15,27)(16,31)(18,32)(19,35)(20,34)(22,37)(23,38)(24,29)(28,33)(36,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5810);