Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#10803 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C7×C8 ◁ G | |
650 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)(9,10)(12,13)(14,18)(15,25,26,19)(16,21)(17,20)(22,34,35,29)(23,31)(24,30)(27,38,39,32)(28,33)(36,42,43,40)(37,41);(8,9,11,10)(12,14,16,23,17,24,28,37)(13,18,21,31,20,30,33,41)(15,22,27,36,26,35,39,43)(19,29,32,40,25,34,38,42);(8,10,11,9)(12,15)(13,19)(14,22)(16,27)(17,26)(18,29)(20,25)(21,32)(23,36)(24,35)(28,39)(30,34)(31,40)(33,38)(37,43)(41,42)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3791);