Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C17 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×#41×C17 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C17 | C22 |
| commutator | C2 | C24×C4×C17 |
| Frattini, cosocle | C22 | C25×C17 |
| socle | C23×C17 | C24 |
| prime radical | C22×#41×C17 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C17 ◁ G | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4,5,6);(7,8)(9,12)(10,11)(13,14);(8,11)(12,14);(7,9,10,13)(8,12,11,14)1360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 2156);