Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C7×#60649 |
| 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 ◁ C4×C7×#41 ◁ G | |
3178 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)(14,22)(15,20)(16,28)(17,19)(18,24)(21,29)(23,35)(25,34)(26,32)(27,40)(30,38)(31,37)(33,39)(36,42)(41,43);(12,14,16,24,17,25,29,38)(13,18,21,22,19,30,28,34)(15,23,27,37,26,36,39,43)(20,31,33,35,32,41,40,42);(8,9,10,11)(13,19)(18,30)(20,32)(21,28)(22,34)(31,41)(33,40)(35,42);(12,15)(13,20)(14,23)(16,27)(17,26)(18,31)(19,32)(21,33)(22,35)(24,37)(25,36)(28,40)(29,39)(30,41)(34,42)(38,43);(15,26)(20,32)(23,36)(27,39)(31,41)(33,40)(35,42)(37,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53637);