Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13046 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| socle | C2×C7 | #1889 |
| prime radical | C7×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,16,13,17,24,30)(10,18,22,32,20,31,35,39)(11,15,19,26,23,29,33,37)(14,25,28,34,27,36,38,21);(8,10,19,25,13,20,33,36)(9,14,26,32,17,27,37,39)(11,21,12,22,23,34,24,35)(15,18,16,28,29,31,30,38);(10,20)(11,23)(12,24)(14,27)(18,31)(25,36)(28,38)(32,39)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6034);