Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #117 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #382 |
| prime radical | C23×C17 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C17×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 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)(20,28,31,24)(21,32,34,47)(22,27)(23,26)(25,33,38,48)(29,42)(30,41,45,37)(35,39)(36,43)(40,49)(44,46);(18,20,22,30,23,31,35,45)(19,24,27,37,26,28,39,41)(21,29,33,43,34,44,48,49)(25,36,32,46,38,40,47,42);(18,21)(19,25,26,38)(20,29)(22,33)(23,34)(24,36,28,40)(27,32,39,47)(30,43)(31,44)(35,48)(37,46,41,42)(45,49)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 565);