Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61887 |
| 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 ◁ C2×C4×C7×Q8 ◁ G | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,15)(12,16)(13,17)(14,20)(19,23)(21,22)(24,25,29,32)(26,33,35,38)(27,31,28,36)(30,37,39,34);(8,10,13,20)(9,14,17,18)(11,19,12,21)(15,22,16,23);(8,11,13,12)(9,15,17,16)(10,19,20,21)(14,22,18,23)(24,26)(25,30)(27,34)(28,37)(29,35)(31,38)(32,39)(33,36);(8,12,13,11)(9,16,17,15)(10,19,20,21)(14,22,18,23);(9,17)(14,18)(15,16)(22,23)(24,27,29,28)(25,31,32,36)(26,34,35,37)(30,38,39,33)427 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54875);