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×#11013 |
| 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 | |
1262 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)(14,22,25,18)(15,19)(16,21)(17,20)(23,40,41,32)(24,38,29,33)(26,43,42,34)(27,36)(28,35)(30,39)(31,37);(12,14)(13,18)(15,26)(16,29)(17,25)(19,34)(20,22)(21,38)(23,27)(24,31)(28,42)(30,41)(32,36)(33,37)(35,43)(39,40);(9,11)(12,15,17,28)(13,19,20,35)(14,23,25,41)(16,30,31,27)(18,32,22,40)(21,39,37,36)(24,42,29,26)(33,43,38,34)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4001);