Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,21,18,14,16,24)(15,20,25,22,26,23,27,19);(12,14,18,13)(15,22,26,19)(16,21,17,24)(20,25,23,27);(9,11)(12,15)(13,19)(14,22)(16,25)(17,27)(18,26)(20,24)(21,23);(12,15)(13,20)(14,23)(16,27)(17,25)(18,26)(19,24)(21,22)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15452);