Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C7×#41 ◁ G | |
4756 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)(8,9)(10,11)(12,13,18,22)(14,23,26,32)(15,20,17,30)(16,21,28,36)(19,31,34,24)(25,37,39,42)(27,35,29,40)(33,41,43,38);(12,14)(13,19)(15,24)(16,25)(17,31)(18,26)(20,32)(21,33)(22,34)(23,30)(27,38)(28,39)(29,41)(35,42)(36,43)(37,40);(9,11)(12,15,18,17)(13,20,22,30)(14,24,26,31)(16,27,28,29)(19,32,34,23)(21,35,36,40)(25,38,39,41)(33,42,43,37);(12,16)(13,21)(14,25)(15,27)(17,29)(18,28)(19,33)(20,35)(22,36)(23,37)(24,38)(26,39)(30,40)(31,41)(32,42)(34,43);(16,28)(21,36)(25,39)(27,29)(33,43)(35,40)(37,42)(38,41)427 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53365);