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×#1418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #357 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(35,36,37,38,39,40,41);(4,9,10,25)(5,13)(6,18)(7,20)(11,29,28,33)(12,30,22,27)(14,19,26,24)(15,21)(16,34)(17,23)(31,32);(3,4)(5,14)(6,19)(7,22)(8,10)(9,23)(11,18)(12,20)(13,25)(15,28)(16,29)(17,26)(21,24)(27,32)(30,34)(31,33)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6609);