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) = C23×C4 |
| direct product | C7×#13043 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C2×C7×B(4,𝔽2) | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,17)(15,22,24,18)(16,19)(20,27)(21,25,23,26);(8,9,10,11)(12,14)(13,17)(15,23)(16,20)(18,26)(19,27)(21,24)(22,25);(13,18)(14,20)(17,25)(19,22)(21,23)(26,27)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6031);