Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #99 |
| commutator | C2×C4 | C42×C17 |
| Frattini, cosocle | C22×Q8 | C22×C17 |
| prime radical | C23×C17 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C17 ◁ C22×Q8×C17 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23,25,29,26,30,34,37)(24,28,31,36,32,27,33,35);(18,19,20,21)(23,27,30,28)(29,36,37,35)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 126);