Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #106 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C2×C17 | C2×#106 |
| prime radical | C23×C17 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 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)(20,29,32,43)(21,26)(22,35)(23,38)(24,28)(25,31,40,45)(27,36)(30,42,33,47)(34,41)(37,48)(39,44,49,46);(18,20,22,31,23,33,36,46,24,32,37,45,21,30,34,44)(19,25,27,29,26,39,35,42,28,40,41,43,38,49,48,47);(18,21,24,23)(19,26,28,38)(20,30,32,33)(22,34,37,36)(25,39,40,49)(27,35,41,48)(29,42,43,47)(31,44,45,46)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 887);