Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13,17,23,32,41,16,20,19,25,15,22,18,24,33,37)(14,26,29,21,35,36,34,43,31,42,27,40,30,38,28,39);(8,9,10,11)(13,20)(15,16)(17,33)(18,32)(21,36)(22,24)(23,41)(25,37)(26,43)(27,34)(28,29)(30,35)(38,40)(39,42);(12,14)(13,21)(15,28)(16,29)(17,34)(18,35)(19,31)(20,36)(22,26)(23,42)(24,43)(25,40)(27,33)(30,32)(37,38)(39,41)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6011);