Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13674 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2196 |
| commutator, Frattini, cosocle | #127 | C23×C7 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×#127 ◁ G | |
454 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 58 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)(11,19)(13,20)(21,22);(8,9,12,15)(10,13,18,22)(11,14,19,23)(16,20,17,21);(8,10)(9,13)(11,16)(12,18)(14,20)(15,22)(17,19)(21,23)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6662);