Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1378 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | C22⋊C4 |
| commutator | C22 | C4×C8×C17 |
| Frattini, cosocle | C22×C8 | C22×C17 |
| socle | C2×C17 | #336 |
| prime radical | C23×C17 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C22×C8×C17 ◁ G | |
132 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 84 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)(18,19,21,25,22,27,32,39,23,26,33,38,34,40,45,48)(20,28,30,41,31,42,44,49,29,24,35,36,43,37,46,47);(18,20)(19,24,26,28)(21,30)(22,31)(23,29)(25,36,38,41)(27,37,40,42)(32,44)(33,35)(34,43)(39,47,48,49)(45,46)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 110);