Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #476 |
| 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)(8,9,12,20,15,21,14,18)(10,22,24,16,25,19,13,30)(11,26,28,37,29,39,27,35)(17,32,33,23,34,31,36,38);(8,10,12,24,15,25,14,13)(9,16,18,22,21,30,20,19)(11,23,27,32,29,38,28,31)(17,26,33,37,34,39,36,35);(8,11)(9,17)(10,23)(12,27)(13,31)(14,28)(15,29)(16,26)(18,33)(19,35)(20,36)(21,34)(22,37)(24,32)(25,38)(30,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5813);