Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C17 · Schur multiplier (H2) = C23 |
| direct product | D4×C16×C17 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C17 | C22 |
| commutator | C2 | C22×C16×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C22×C17 | C22×C8 |
| prime radical | C2×D4×C17 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C17 ◁ G | |
174 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 174 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) = 37 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(2,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)1360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 899);