Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C7×#60651 |
| 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 | |
3138 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)(12,13,17,21)(14,22,23,35)(15,19,26,33)(16,28,29,20)(18,30,31,25)(24,36,37,43)(27,40,39,34)(32,41,42,38);(12,14,16,25,17,23,29,30)(13,18,20,22,21,31,28,35)(15,24,27,38,26,37,39,41)(19,32,34,36,33,42,40,43);(8,9,10,11)(14,23)(18,31)(22,35)(24,37)(25,30)(32,42)(36,43)(38,41);(12,15)(13,19)(14,24)(16,27)(17,26)(18,32)(20,34)(21,33)(22,36)(23,37)(25,38)(28,40)(29,39)(30,41)(31,42)(35,43);(15,26)(19,33)(24,37)(27,39)(32,42)(34,40)(36,43)(38,41)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53639);