Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#11015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1242 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)(10,11)(12,13,17,23,18,21,16,22)(14,24,27,34,28,19,26,35)(15,20,30,38,31,37,32,39)(25,40,29,43,41,33,42,36);(12,14,18,28)(13,19,21,24)(15,29,31,42)(16,27,17,26)(20,36,37,43)(22,35,23,34)(25,32,41,30)(33,39,40,38);(9,11)(12,15)(13,20)(14,25)(16,32)(17,30)(18,31)(19,33)(21,37)(22,39)(23,38)(24,40)(26,42)(27,29)(28,41)(34,43)(35,36)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4003);