Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#1684 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C17 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C17 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C42×C17 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 2176 |
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)(18,19,22,25)(20,26,29,33)(21,30,27,24)(23,31,32,28)(34,35)(36,42)(37,39)(38,46)(40,44)(41,45)(43,49)(47,48);(18,20,27,31,22,29,21,28)(19,23,30,33,25,32,24,26)(35,39)(40,48)(41,46)(42,49);(34,36,37,43)(35,40,39,48)(38,47,45,44)(41,49,46,42)493 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 416);