Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | D8 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C2×C17 | C4×D8 |
| prime radical | C17×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C2×C8×C17 ◁ C4×C8×C17 ◁ G | |
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,23,27,24,29,39,28)(20,30,32,41,34,47,33,49)(21,26,35,44,36,45,22,37)(25,40,42,48,43,38,46,31);(18,20)(19,25)(21,31)(22,38)(23,32)(24,34)(26,41)(27,42)(28,46)(29,43)(30,37)(33,39)(35,40)(36,48)(44,47)(45,49);(18,21,23,35,24,36,39,22)(19,26,27,44,29,45,28,37)(20,31,32,40,34,48,33,38)(25,41,42,47,43,49,46,30)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 910);