Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1946 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C22×C17 | C4×D4 |
| prime radical | C23×C17 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C8×C17 ◁ G | |
200 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 200 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)(22,23,26,31,27,32,36,30)(24,28,25,34,33,29,35,37);(19,21)(22,24,26,25,27,33,36,35)(23,28,31,34,32,29,30,37);(22,25,26,33,27,35,36,24)(23,29,30,34,32,28,31,37)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 678);