Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4×C17 |
| Frattini, cosocle | #124 | C22×C17 |
| prime radical | C17×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C17 ◁ C2×C4×C17 ◁ C17×#124 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,24)(21,30,31,27)(22,26,29,33)(23,32,25,28);(18,19)(20,27)(21,30)(22,26)(23,29)(24,28)(25,32)(31,33)289 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 136);