Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13170 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
278 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)(12,13,17,22,18,24,33,23)(14,19,16,32,26,21,34,36)(15,27,30,41,31,35,40,42)(20,28,37,43,39,25,38,29);(9,11)(12,14,17,16,18,26,33,34)(13,19,22,32,24,21,23,36)(15,28,30,43,31,25,40,29)(20,27,37,41,39,35,38,42);(12,15,26,28,17,30,34,43,18,31,14,25,33,40,16,29)(13,20,32,42,23,38,19,35,24,39,36,41,22,37,21,27)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6158);