Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #99 |
| commutator | C23 | C2×C8×C17 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C17 |
| prime radical | C24×C17 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C22⋊C4×C17 ◁ G | |
150 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,22,21,23,24,25)(26,27)(28,34)(29,37,38,32)(30,33)(31,35,39,41)(36,40);(26,28,30,36)(27,31,33,39)(29,35,38,41)(32,34,37,40)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 49);