Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C7×#60643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
3194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,23)(15,20)(16,21)(17,31)(18,22)(19,26)(24,36)(25,33)(27,37)(28,34)(29,35)(30,41)(32,40)(38,43)(39,42);(12,14,17,26,18,27,16,25)(13,19,21,23,22,33,31,37)(15,24,30,40,28,38,29,39)(20,32,35,36,34,42,41,43);(8,9,10,11)(12,15)(13,20)(14,24)(16,29)(17,30)(18,28)(19,32)(21,35)(22,34)(23,36)(25,39)(26,40)(27,38)(31,41)(33,42)(37,43);(15,28)(20,34)(24,38)(29,30)(32,42)(35,41)(36,43)(39,40);(12,16,18,17)(13,21,22,31)(14,25,27,26)(15,29,28,30)(19,23,33,37)(20,35,34,41)(24,39,38,40)(32,36,42,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53631);