Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#12298 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23×C7 |
| prime radical | C7×#3192 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
376 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,13,16,12,17,11,22)(10,18,19,23,20,14,21,15);(9,14)(10,19)(11,13)(15,22)(16,23)(17,18)(20,21);(10,20)(19,21)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5286);