Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7193 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #99 |
| commutator | C2×C4 | C4×C7×C8 |
| Frattini, cosocle | C22×M16 | C22×C7 |
| socle | C22×C7 | #324 |
| prime radical | C24×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C22×C7×M16 ◁ G | |
342 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(12,13,15,20,17,18,23,25,16,21,24,27,14,19,22,26);(8,9,10,11)(13,18,21,19)(20,26,27,25)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 181);