Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11272 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #389 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13,17,21,18,23,16,22)(14,24,27,35,28,38,26,19)(15,29,32,37,33,39,31,20)(25,41,42,36,43,34,30,40);(8,10,11,9)(12,14,16,26,18,28,17,27)(13,19,21,24,23,35,22,38)(15,30,31,43,33,42,32,25)(20,36,29,34,37,40,39,41);(9,10)(12,15,16,31,18,33,17,32)(13,20,21,29,23,37,22,39)(14,25,26,30,28,43,27,42)(19,34,24,40,35,41,38,36)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4260);