Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33561 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C7 |
| prime radical | C7×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
1096 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)(21,32)(23,30)(24,29)(25,33)(26,34)(27,35)(28,36)(31,37)(38,39);(8,10,14,11)(9,15,19,29)(12,23,26,21)(13,22,27,24)(16,18,20,35)(17,31,34,28)(25,37,38,36)(30,39,32,33);(8,11,14,10)(9,16,19,20)(12,21,26,23)(13,22,27,24)(15,18,29,35)(17,32,34,30)(25,37,38,36)(28,39,31,33);(8,12,14,26)(9,17,19,34)(10,21,11,23)(13,25,27,38)(15,28,29,31)(16,30,20,32)(18,33,35,39)(22,36,24,37)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26549);