Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #459 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
476 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)(12,13,18,25,38,39,37,23,20,27,17,24,19,26,16,36)(14,28,31,21,35,22,29,42,33,43,30,41,32,40,15,34);(12,14,19,35,20,33,38,32)(13,21,39,34,27,41,26,42)(15,16,31,17,29,37,30,18)(22,25,28,23,40,24,43,36);(9,11)(12,15)(13,22)(14,16)(17,35)(18,32)(19,31)(20,29)(21,25)(23,34)(24,41)(26,43)(27,40)(28,39)(30,38)(33,37)(36,42)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6016);