Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13153 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #442 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,18,14,22)(15,19)(16,25)(20,23)(21,27,24,26);(12,13,19,27,25,24,15,22,17,14,23,26,16,21,20,18);(9,11)(12,14,15,22,16,24,19,27,17,13,20,18,25,21,23,26)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6141);