Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#12850 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #117 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | #432 |
| prime radical | C24×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#97 ◁ G | |
430 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)(12,13,16,23,17,20,32,36,18,21,31,37,15,22,29,38)(14,24,27,39,28,33,41,42,25,19,30,34,26,35,40,43);(12,14)(13,19)(15,26)(16,30)(17,28)(18,25)(20,35)(21,24)(22,33)(23,39)(27,31)(29,41)(32,40)(34,37)(36,42)(38,43);(9,11)(13,20,21,22)(14,25)(16,31)(19,33,24,35)(23,38,37,36)(26,28)(29,32)(34,43,39,42)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5838);