Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#1899 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C17 |
| Frattini, cosocle | C24 | C23×C17 |
| prime radical | C22×D4×C17 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G | |
378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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)(24,27,31,34)(25,28)(26,29)(30,33,36,37)(32,35);(22,24)(23,27,29,34)(25,30)(26,31)(28,33,35,37)(32,36);(19,21)(22,25,26,32)(23,28,29,35)(24,30,31,36)(27,33,34,37)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 631);