Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #99 |
| commutator | C23 | C42×C17 |
| Frattini, cosocle | C2×#41 | C22×C17 |
| socle | C2×C17 | #342 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,28)(21,32,33,25)(22,26)(23,27)(24,29,36,42)(30,40)(31,41)(34,45,46,38)(35,39)(37,43,47,49)(44,48);(18,20,22,30,23,31,35,44)(19,24,26,37,27,36,39,47)(21,29,34,43,33,42,46,49)(25,28,38,40,32,41,45,48)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 123);