Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C23×C7 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13,17,22,19,23,18,33)(14,24,26,37,27,36,32,20)(15,28,29,38,30,41,16,21)(25,34,31,39,40,43,42,35);(8,10,11,9)(12,14,19,27)(13,20,23,37)(15,25,30,40)(16,31,29,42)(17,32,18,26)(21,34,38,43)(22,36,33,24)(28,35,41,39);(12,15,17,29,19,30,18,16)(13,21,33,41,23,38,22,28)(14,25,32,42,27,40,26,31)(20,34,24,39,37,43,36,35);(12,16,18,30,19,29,17,15)(13,21,33,41,23,38,22,28)(14,25,32,42,27,40,26,31)(20,35,36,43,37,39,24,34)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16188);