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×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C22×C7×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
516 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)(9,11)(12,13,15,20,30,39,16,32,18,23,31,40,17,22,29,21)(14,24,25,41,34,36,43,38,28,42,33,37,27,35,26,19);(8,9,10,11)(12,14)(13,19)(15,26)(16,33)(17,34)(18,28)(20,35)(21,24)(22,41)(23,38)(25,29)(27,30)(31,43)(32,42)(36,40)(37,39);(13,20)(15,29)(16,31)(17,30)(19,35)(21,39)(22,32)(23,40)(24,37)(25,26)(27,34)(33,43)(36,38)(41,42)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6005);