Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C7 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,20)(11,16)(12,17)(13,18)(14,19)(15,22)(21,34)(23,35)(24,30)(25,39)(26,32)(27,33)(28,36)(29,38)(31,37);(8,10)(9,15)(11,21)(12,25)(13,22)(14,23)(16,28)(17,31)(18,20)(19,29)(24,36)(26,37)(27,38)(30,34)(32,39)(33,35);(8,11,14,12)(9,16,19,17)(10,21,23,25)(13,26,27,24)(15,28,29,31)(18,30,33,32)(20,34,35,39)(22,37,38,36);(8,12,14,11)(9,17,19,16)(10,21,23,25)(13,26,27,24)(15,28,29,31)(18,32,33,30)(20,34,35,39)(22,36,38,37)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26543);