Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#1784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C24×C17 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G | |
564 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,25,29,26,28,34,36)(24,30,31,37,32,27,33,35);(22,24,26,32)(23,27,28,30)(25,33,34,31)(29,37,36,35);(19,21)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 516);