Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C22×C17 | C2×C22⋊C4 |
| prime radical | C23×C17 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,24)(21,22)(23,25)(27,30,31,32);(19,22)(23,24)(26,27,28,32,29,31,33,30);(18,20,21,25)(19,23,22,24)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 718);