Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11297 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #385 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
578 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)(12,13,17,22,18,23,16,21)(14,24,27,19,28,34,26,35)(15,29,30,20,32,36,31,37)(25,39,40,43,41,33,42,38);(8,9)(10,11)(12,14,16,27,18,28,17,26)(13,19,21,34,23,35,22,24)(15,25,31,40,32,41,30,42)(20,33,29,38,37,39,36,43);(9,11)(12,15,16,30,18,32,17,31)(13,20,21,36,23,37,22,29)(14,25,26,40,28,41,27,42)(19,33,24,38,35,39,34,43)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4285);