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×#13022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
494 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)(12,13,15,21,32,36,16,22,18,24,33,37,17,23,30,19)(14,25,26,20,34,38,31,42,29,41,43,40,28,35,27,39);(8,9,10,11)(13,19)(15,30)(16,33)(17,32)(20,35)(21,23)(22,24)(25,39)(26,27)(28,34)(31,43)(36,37)(38,40)(41,42);(12,14)(13,20)(15,31)(16,26)(17,34)(18,29)(19,35)(21,41)(22,38)(23,42)(24,40)(25,37)(27,33)(28,32)(30,43)(36,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6010);