Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#1348 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #102 |
| commutator | C2×C8 | C2×C4×C17 |
| Frattini, cosocle | C22×C8 | C22×C17 |
| prime radical | C23×C17 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G | |
116 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,24,25,22)(21,23)(27,31)(28,34)(29,36)(32,41)(33,40)(35,38)(37,39);(18,20)(19,22)(21,25)(23,24)(26,27,30,33)(28,35,37,41)(29,38,36,32)(31,39,40,34)493 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 80);