Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#1540 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C42 | C23×C17 |
| socle | C22×C17 | C2×C22⋊C4 |
| prime radical | C17×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ G | |
256 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 1088 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,23,22,25,20,24)(26,27,30,35,31,34,29,33)(28,36,37,41,38,32,39,40);(19,23)(20,21)(24,25)(26,28)(27,32)(29,37)(30,39)(31,38)(33,40)(34,36)(35,41);(26,29,31,30)(27,33,34,35)(28,37,38,39)(32,40,36,41)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 272);