Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | C22⋊C4 |
| commutator | C22 | C2×C42×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | C4×C22⋊C4 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C22×C8×C17 ◁ G | |
284 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,26,23,27,34,40)(20,28,31,42,30,24,35,37)(21,25,32,38,33,39,45,48)(29,41,44,49,43,36,46,47);(18,20)(19,24,27,28)(21,29)(22,31)(23,30)(25,36,39,41)(26,37,40,42)(32,44)(33,43)(34,35)(38,47,48,49)(45,46);(18,21,22,32,23,33,34,45)(19,25,26,38,27,39,40,48)(20,29,31,44,30,43,35,46)(24,36,37,47,28,41,42,49)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 489);