Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#1979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #121 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C23×C17 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ G | |
276 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,21)(22,23,27,32,28,31,26,40)(24,29,35,44,34,45,41,43)(25,36,38,46,39,50,37,30)(33,47,49,42,48,52,51,53);(18,19,20,21)(22,24,26,35,28,34,27,41)(23,29,32,43,31,45,40,44)(25,33,37,49,39,48,38,51)(30,42,36,47,46,53,50,52);(22,25,27,38,28,39,26,37)(23,30,40,50,31,46,32,36)(24,33,41,51,34,48,35,49)(29,42,44,52,45,53,43,47)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 711);