Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#1420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #357 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| prime radical | C23×C7 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2)(67,68,69,70,71,72,73);(3,4,7,13,10,15,9,33)(5,16,18,47,21,49,20,56)(6,22,25,57,28,59,27,64)(8,29,24,45,32,40,31,14)(11,34,35,17,37,51,43,53)(12,38,41,63,42,62,58,26)(19,54,50,39,55,66,30,48)(23,44,60,65,52,46,61,36);(3,5,10,21)(4,11,15,37)(6,23,28,52)(7,20,9,18)(8,30,32,50)(12,39,42,48)(13,43,33,35)(14,44,45,46)(16,22,49,59)(17,26,53,63)(19,24,55,31)(25,61,27,60)(29,36,40,65)(34,38,51,62)(41,54,58,66)(47,64,56,57)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6612);