Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #121 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #389 |
| prime radical | C23×C17 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ G | |
184 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 108 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)(18,19,23,27,24,28,22,38)(20,29,31,25,33,40,32,41)(21,34,36,42,37,48,35,26)(30,39,44,47,46,49,45,43);(18,20,23,31,24,33,22,32)(19,25,27,40,28,41,38,29)(21,30,36,44,37,46,35,45)(26,39,34,47,42,49,48,43);(18,21,22,35,24,37,23,36)(19,26,27,34,28,42,38,48)(20,30,32,45,33,46,31,44)(25,39,40,47,41,49,29,43)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 685);