Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C26 |
| direct product | C17×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C17 |
| Frattini, cosocle | C2×C4 | C24×C17 |
| prime radical | C2×C17×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×#41×C17 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,27,23,26,35,40)(20,28,30,41,31,24,34,37)(21,25,33,39,32,38,45,48)(29,42,44,49,43,36,46,47);(18,20,23,31)(19,24,26,28)(21,29,32,43)(22,34,35,30)(25,36,38,42)(27,41,40,37)(33,46,45,44)(39,49,48,47);(18,21)(19,25)(20,29)(22,33)(23,32)(24,36)(26,38)(27,39)(28,42)(30,44)(31,43)(34,46)(35,45)(37,47)(40,48)(41,49);(21,32)(25,38)(29,43)(33,45)(36,42)(39,48)(44,46)(47,49)697 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 1621);